FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS — THE LECTURE NOTES FOR COURSE (189) 261/325
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FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS — THE LECTURE NOTES FOR COURSE (189) 261/325

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

JIAN-JUN XU AND JOHN LABUTE Department of Mathematics and Statistics, McGill University

Kluwer Academic Publishers Boston/Dordrecht/London

Contents

1. INTRODUCTION 1 Definitions and Basic Concepts 1.1 Ordinary Differential Equation (ODE) 1.2 Solution 1.3 Order n of the DE 1.4 Linear Equation: 1.5 Homogeneous Linear Equation: 1.6 Partial Differential Equation (PDE) 1.7 General Solution of a Linear Differential Equation 1.8 A System of ODE’s 2 The Approaches of Finding Solutions of ODE 2.1 Analytical Approaches 2.2 Numerical Approaches 2. FIRST ORDER DIFFERENTIAL EQUATIONS 1 Linear Equation 1.1 Linear homogeneous equation 1.2 Linear inhomogeneous equation 2 Separable Equations. 3 Logistic Equation 4 Fundamental Existence and Uniqueness Theorem 5 Bernoulli Equation: 6 Homogeneous Equation: 7 Exact Equations. 8 Theorem. 9 Integrating Factors. v

1 1 1 1 1 2 2 2 3 3 4 4 4 5 7 7 8 11 13 14 15 16 19 20 21

vi

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

10

11 12 13 14 15 16

17 18 19 20 21

Change of Variables. 10.1 y 0 = f (ax + by), b 6= 0 dy a1 x + b1 y + c1 10.2 = dx a2 x + b2 y + c2 10.3 Riccatti equation: y 0 = p(x)y + q(x)y 2 + r(x) Orthogonal Trajectories. Falling Bodies with Air Resistance Mixing Problems Heating and Cooling Problems Radioactive Decay Definitions and Basic Concepts 16.1 Directional Field 16.2 Integral Curves 16.3 Autonomous Systems 16.4 Equilibrium Points Phase Line Analysis Bifurcation Diagram Euler’s Method Improved Euler’s Method Higher Order Methods

3. N-TH ORDER DIFFERENTIAL EQUATIONS 1 Theorem of Existence and Uniqueness (I) 1.1 Lemma 2 Theorem of Existence and Uniqueness (II) 3 Theorem of Existence and Uniqueness (III) 3.1 Case (I) 3.2 Case (II) 4 Linear Equations 4.1 Basic Concepts and General Properties 5 Basic Theory of Linear Differential Equations 5.1 Basics of Linear Vector Space 5.1.1 Isomorphic Linear Transformation 5.1.2 Dimension and Basis of Vector Space 5.1.3 (*) Span and Subspace 5.1.4 Linear Independency 5.2 Wronskian of n-functions

23 23 23 24 25 27 27 28 29 31 31 31 31 31 32 32 37 38 38 43 46 46 47 47 49 50 50 50 51 51 51 52 52 52 53

vii

Contents

6

7

8 9 10 11 12 13

5.2.1 Definition 5.2.2 Theorem 1 5.2.3 Theorem 2 The Method with Undetermined Parameters 6.1 Basic Equalities (I) 6.2 Cases (I) ( r1 > r2 ) 6.3 Cases (II) ( r1 = r2 ) 6.4 Cases (III) ( r1,2 = λ ± iµ) The Method with Differential Operator 7.1 Basic Equalities (II). 7.2 Cases (I) ( b2 − 4ac > 0) 7.3 Cases (II) ( b2 − 4ac = 0) 7.4 Cases (III) ( b2 − 4ac < 0) 7.5 Theorems The Differential Operator for Equations with Constant Coefficients The Method of Variation of Parameters Euler Equations Exact Equations Reduction of Order (*) Vibration System

4. SERIES SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS 1 Series Solutions near a Ordinary Point 1.1 Theorem 2 Series Solutions near a Regular Singular Point 2.1 Case (I): The roots (r1 − r2 6= N ) 2.2 Case (II): The roots (r1 = r2 ) 2.3 Case (III): The roots (r1 − r2 = N > 0) 3 Bessel Equation 4 The Case of Non-integer ν 5 The Case of ν = −m with m an integer ≥ 0 5. LAPLACE TRANSFORMS 1 Introduction 2 Laplace Transform 2.1 Definition

53 54 54 57 57 58 59 60 61 61 62 62 63 64 67 68 71 73 74 77 81 84 84 87 89 89 90 95 96 96 101 103 104 104

viii

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

3

4

5

6

7

2.2 Basic Properties and Formulas 2.2.1 Linearity of the transform 2.2.2 Formula (I) 2.2.3 Formula (II) 2.2.4 Formula (III) Inverse Laplace Transform 3.1 Theorem: 3.2 Definition Solve IVP of DE’s with Laplace Transform Method 4.1 Example 1 4.2 Example 2 Step Function 5.1 Definition 5.2 Laplace transform of unit step function Impulse Function 6.1 Definition 6.2 Laplace transform of unit step function Convolution Integral 7.1 Theorem

105 105 106 106 106 107 107 107 109 109 111 113 113 113 113 113 114 114 114

6. (*) SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS 1 Mathematical Formulation of a Practical Problem 2 (2 × 2) System of Linear Equations 2.1 Case 1: ∆ > 0 2.2 Case 2: ∆ < 0 2.3 Case 3: ∆ = 0

115 117 119 119 120 121

Appendices ASSIGNMENTS AND SOLUTIONS

127 127

Chapter 1 INTRODUCTION

1. 1.1

Definitions and Basic Concepts Ordinary Differential Equation (ODE)

An equation involving the derivatives of an unknown function y of a single variable x over an interval x ∈ (I).

1.2

Solution

Any function y = f (x) which satisfies this equation over the interval (I) is called a solution of the ODE. For example, y = e2x is a solution of the ODE y 0 = 2y and y = sin(x2 ) is a solution of the ODE xy 00 − y 0 + 4x3 y = 0.

1.3

Order n of the DE

An ODE is said to be order n, if y (n) is the highest order derivative occurring in the equation. The simplest first order ODE is y 0 = g(x). The most general form of an n-th order ODE is F (x, y, y 0 , . . . , y (n) ) = 0 with F a function of n + 2 variables x, u0 , u1 , . . . , un . The equations xy 00 + y = x3 ,

y 0 + y 2 = 0,

1

y 000 + 2y 0 + y = 0

2

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

are examples of ODE’s of second order, first order and third order respectively with respectively F (x, u0 , u1 , u2 ) = xu2 + u0 − x3 , F (x, u0 , u1 ) = u1 + u20 , F (x, u0 , u1 , u2 , u3 ) = u3 + 2u1 + u0 .

1.4

Linear Equation:

If the function F is linear in the variables u0 , u1 , . . . , un the ODE is said to be linear. If, in addition, F is homogeneous then the ODE is said to be homogeneous. The first of the above examples above is linear are linear, the second is non-linear and the third is linear and homogeneous. The general n-th order linear ODE can be written an (x)

1.5

dn y dn−1 y dy + a0 (x)y = b(x). + a (x) + · · · + a1 (x) n−1 n n−1 dx dx dx

Homogeneous Linear Equation:

The linear DE is homogeneous, if and only if b(x) ≡ 0. Linear homogeneous equations have the important property that linear combinations of solutions are also solutions. In other words, if y1 , y2 , . . . , ym are solutions and c1 , c2 , . . . , cm are constants then c1 y1 + c2 y2 + · · · + cm ym is also a solution.

1.6

Partial Differential Equation (PDE)

An equation involving the partial derivatives of a function of more than one variable is called PED. The concepts of linearity and homogeneity can be extended to PDE’s. The general second order linear PDE in two variables x, y is a(x, y)

∂2u ∂2u ∂2u ∂u + c(x, y) + b(x, y) + d(x, y) 2 2 ∂x ∂x∂y ∂y ∂x ∂u +e(x, y) + f (x, y)u = g(x, y). ∂y

Laplace’s equation

∂2u ∂2u + 2 =0 ∂x2 ∂y

is a linear, homogeneous PDE of order 2. The functions u = log(x2 +y 2 ), u = xy, u = x2 − y 2 are examples of solutions of Laplace’s equation. We will not study PDE’s systematically in this course.

3

INTRODUCTION

1.7

General Solution of a Linear Differential Equation

It represents the set of all solutions, i.e., the set of all functions which satisfy the equation in the interval (I). For example, the general solution of the differential equation y 0 = 3x2 is y = x3 + C where C is an arbitrary constant. The constant C is the value of y at x = 0. This initial condition completely determines the solution. More generally, one easily shows that given a, b there is a unique solution y of the differential equation with y(a) = b. Geometrically, this means that the one-parameter family of curves y = x2 + C do not intersect one another and they fill up the plane R2 .

1.8

A System of ODE’s y10 = G1 (x, y1 , y2 , . . . , yn ) y20 = G2 (x, y1 , y2 , . . . , yn ) .. . yn0 = Gn (x, y1 , y2 , . . . , yn )

An n-th order ODE of the form y (n) = G(x, y, y 0 , . . . , y n−1 ) can be transformed in the form of the system of first order DE’s. If we introduce dependant variables y1 = y, y2 = y 0 , . . . , yn = y n−1 we obtain the equivalent system of first order equations y10 = y2 , y20 = y3 , .. .

(1.1)

yn0 = G(x, y1 , y2 , . . . , yn ). For example, the ODE y 00 = y is equivalent to the system y10 = y2 , y20 = y1 .

(1.2)

In this way the study of n-th order equations can be reduced to the study of systems of first order equations. Some times, one called the latter as the normal form of the n-th order ODE. Systems of equations arise in the study of the motion of particles. For example, if P (x, y) is the position of a particle of mass m at time t, moving in a plane under the action of the force field (f (x, y), g(x, y), we have 2

m ddt2x = f (x, y), 2 m ddt2y = g(x, y).

(1.3)

4

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

This is a second order system. The general first order ODE in normal form is y 0 = F (x, y). If F and ∂F ∂y are continuous one can show that, given a, b, there is a unique solution with y(a) = b. Describing this solution is not an easy task and there are a variety of ways to do this. The dependence of the solution on initial conditions is also an important question as the initial values may be only known approximately. The non-linear ODE yy 0 = 4x is not in normal form but can be brought to normal form 4x y0 = . y by dividing both sides by y.

2. 2.1

The Approaches of Finding Solutions of ODE Analytical Approaches

Analytical solution methods: finding the exact form of solutions; Geometrical methods: finding the qualitative behavior of solutions; Asymptotic methods: finding the asymptotic form of the solution, which gives good approximation of the exact solution.

2.2

Numerical Approaches

Numerical algorithms — numerical methods; Symbolic manipulators — Maple, MATHEMATICA, MacSyma. This course mainly discuss the analytical approaches and mainly on analytical solution methods.

Chapter 2 FIRST ORDER DIFFERENTIAL EQUATIONS

5

PART (I): LINEAR EQUATIONS

In this lecture we will treat linear and separable first order ODE’s.

1.

Linear Equation

The general first order ODE has the form F (x, y, y 0 ) = 0 where y = y(x). If it is linear it can be written in the form a0 (x)y 0 + a1 (x)y = b(x) where a0 (x), a( x), b(x) are continuous functions of x on some interval (I). To bring it to normal form y 0 = f (x, y) we have to divide both sides of the equation by a0 (x). This is possible only for those x where a0 (x) 6= 0. After possibly shrinking I we assume that a0 (x) 6= 0 on (I). So our equation has the form (standard form) y 0 + p(x)y = q(x) with p(x) = a1 (x)/a0 (x) and q(x) = b(x)/a0 (x), both continuous on (I). Solving for y 0 we get the normal form for a linear first order ODE, namely y 0 = q(x) − p(x)y.

1.1

Linear homogeneous equation

Let us first consider the simple case: q(x) = 0, namely, dy + p(x)y = 0. dx With the chain law of derivative, one may write d y 0 (x) = ln [y(x)] = −p(x), y dx

7

8

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

integrating both sides, we derive Z

ln y(x) = − or

p(x)dx + C,

y = C1 e−

where C, as well as C1 =

1.2

eC ,

R

p(x)dx

,

is arbitrary constant.

Linear inhomogeneous equation

We now consider the general case: dy + p(x)y = q(x). dx We multiply the both sides of our differential equation with a factor µ(x) 6= 0. Then our equation is equivalent (has the same solutions) to the equation µ(x)y 0 (x) + µ(x)p(x)y(x) = µ(x)q(x). We wish that with a properly chosen function µ(x), d [µ(x)y(x)]. dx

µ(x)y 0 (x) + µ(x)p(x)y(x) =

For this purpose, the function µ(x) must has the property µ0 (x) = p(x)µ(x),

(2.1)

and µ(x) 6= 0 for all x. By solving the linear homogeneous equation (2.1), one obtain R

µ(x) = e

p(x)dx

.

(2.2)

With this function, which is called an integrating factor, our equation is reduced to d [µ(x)y(x)] = µ(x)q(x), dx

(2.3)

Integrating both sides, we get Z

µ(x)y =

µ(x)q(x)dx + C

with C an arbitrary constant. Solving for y, we get y=

1 µ(x)

Z

µ(x)q(x)dx +

C = yP (x) + yH (x) µ(x)

(2.4)

9

FIRST ORDER DIFFERENTIAL EQUATIONS

as the general solution for the general linear first order ODE y 0 + p(x)y = q(x). In solution (2.4), the first part, yP (x), is a particular solution of the inhomogeneous equation, while the second part, yH (x), is the general solution of the associate homogeneous solution. Note that for any pair of scalars a, b with a in (I), there is a unique scalar C such that y(a) = b. Geometrically, this means that the solution curves y = φ(x) are a family of non-intersecting curves which fill the region I × R. Example 1: y 0 + xy = x. This is a linear first order ODE in standard form with p(x) = q(x) = x. The integrating factor is R

xdx

µ(x) = e

2 /2

= ex

.

Hence, after multiplying both sides of our differential equation, we get d x2 /2 2 (e y) = xex /2 dx which, after integrating both sides, yields 2 /2

ex

Z

2 /2

xex

y=

dx + C = ex

2 /2

+ C.

2

Hence the general solution is y = 1+Ce−x /2 . The solution satisfying the initial condition y(0) = 1 is y = 1 and the solution satisfying y(0) = a 2 is y = 1 + (a − 1)e−x /2 . Example 2: xy 0 − 2y = x3 sin x, (x > 0). We bring this linear first order equation to standard form by dividing by x. We get y0 + The integrating factor is µ(x) = e

R

−2 y = x2 sin x. x

−2dx/x

= e−2 ln x = 1/x2 .

After multiplying our DE in standard form by 1/x2 and simplifying, we get d (y/x2 ) = sin x dx from which y/x2 = − cos x + C and y = −x2 cos x + Cx2 . Note that the later are solutions to the DE xy 0 − 2y = x3 sin x and that they all satisfy the initial condition y(0) = 0. This non-uniqueness is due to the fact that x = 0 is a singular point of the DE.

PART (II): SEPARABLE EQUATIONS — NONLINEAR EQUATIONS (1)

2.

Separable Equations.

The first order ODE y 0 = f (x, y) is said to be separable if f (x, y) can be expressed as a product of a function of x times a function of y. The DE then has the form y 0 = g(x)h(y) and, dividing both sides by h(y), it becomes y0 = g(x). h(y) Of course this is not valid for those solutions y = y(x) at the points where φ(x) = 0. Assuming the continuity of g and h, we can integrate both sides of the equation to get Z

y 0 (x) dx = h[y(x)]

Assume that

Z

g(x)dx + C. Z

H(y) =

dy , h(y)

By chain rule, we have d 1 H[y(x)] = H 0 (y)y 0 (x) = y 0 (x), dx h[y(x)] hence

Z

H[y(x)] = Therefore,

Z

y 0 (x) dx = h[y(x)]

dy = H(y) = h(y)

11

Z

g(x)dx + C.

Z

g(x)dx + C,

12

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

gives the implicit form of the solution. It determines the value of y implicitly in terms of x. Example 1: y 0 = x−5 . y2 To solve it using the above method we multiply both sides of the equation by y 2 to get y 2 y 0 = (x − 5). Integrating both sides we get y 3 /3 = x2 /2 − 5x + C. Hence, h

y = 3x2 /2 − 15x + C1

i1/3

.

y−1 Example 2: y 0 = x+3 (x > −3). By inspection, y = 1 is a solution. Dividing both sides of the given DE by y − 1 we get

1 y0 = . y−1 x+3 This will be possible for those x where y(x) 6= 1. Integrating both sides we get Z Z dx y0 dx = + C1 , y−1 x+3 from which we get ln |y − 1| = ln(x + 3) + C1 . Thus |y − 1| = eC1 (x + 3) from which y − 1 = ±eC1 (x + 3). If we let C = ±eC1 , we get y = 1 + C(x + 3) which is a family of lines passing through (−3, 1); for any (a, b) with b 6= 0 there is only one member of this family which passes through (a, b). Since y = 1 was found to be a solution by inspection the general solution is y = 1 + C(x + 3), where C can be any scalar. y cos x Example 3: y 0 = 1+2y Transforming in the standard form then 2. integrating both sides we get

Z

(1 + 2y 2 ) dy = y

Z

cos x dx + C,

from which we get a family of the solutions: ln |y| + y 2 = sin x + C,

FIRST ORDER DIFFERENTIAL EQUATIONS

13

where C is an arbitrary constant. However, this is not the general solution of the equation, as it does not contains, for instance, the solution: y = 0. With I.C.: y(0)=1, we get C = 1, hence, the solution: ln |y| + y 2 = sin x + 1.

3.

Logistic Equation y 0 = ay(b − y),

where a, b > 0 are fixed constants. This equation arises in the study of the growth of certain populations. Since the right-hand side of the equation is zero for y = 0 and y = b, the given DE has y = 0 and y = b as solutions. More generally, if y 0 = f (t, y) and f (t, c) = 0 for all t in some interval (I), the constant function y = c on (I) is a solution of y 0 = f (t, y) since y 0 = 0 for a constant function y. To solve the logistic equation, we write it in the form y0 = a. y(b − y) Integrating both sides with respect to t we get Z

y 0 dt = at + C y(b − y)

which can, since y 0 dt = dy, be written as Z

dy = at + C. y(b − y)

Since, by partial fractions, 1 1 1 1 = ( + ) y(b − y) b y b−y we obtain

1 (ln |y| − ln |b − y|) = at + C. b Multiplying both sides by b and exponentiating both sides to the base e, we get |y| = ebC eabt = C1 eabt , |b − y| where the arbitrary constant C1 = ±ebC can be determined by the initial condition (IC): y(0) = y0 as C1 =

|y0 | . |b − y0 |

14

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

Two cases need to be discussed separately. y0 Case (I), y0 < b: one has C1 = | b−y |= 0

|y| = |b − y|

µ

y0 b−y0

> 0. So that,

¶

y0 eabt > 0, b − y0

(t ∈ (I)).

From the above we derive y/(b − y) = C1 eabt , and y = (b − y)C1 eabt . This gives ³ ´ y0 b b−y eabt bC1 eabt 0 ³ ´ y= = . 1 + C1 eabt 1 + y0 eabt b−y0

It shows that if y0 = 0, one has the solution y(t) = 0. However, if 0 < y0 < b, one has the solution 0 < y(t) < b, and as t → ∞, y(t) → b. y0 y0 Case (II), y0 > b: one has C1 = | b−y | = − b−y > 0. So that, 0 0

¯ µ ¯ ¶ ¯ y ¯ y0 ¯= ¯ eabt > 0, (t ∈ (I)). ¯b − y ¯ y0 − b ´ ³

From the above we derive y/(y − b) = ³

b)

y0 y0 −b

´

y0 y0 −b

eabt , and y = (y −

eabt . This gives ³

y=³

b

y0 y0 −b

y0 y0 −b

´

´

eabt

eabt − 1

.

It shows that if y0 > b, one has the solution y(t) > b, and as t → ∞, y(t) → b. It is derived that y(t) = 0 is an unstable equilibrium state of the system; y(t) = b is a stable equilibrium state of the system.

4.

Fundamental Existence and Uniqueness Theorem

If the function f (x, y) together with its partial derivative with respect to y are continuous on the rectangle R : |x − x0 | ≤ a, |y − y0 | ≤ b there is a unique solution to the initial value problem y 0 = f (x, y),

y(x0 ) = y0

FIRST ORDER DIFFERENTIAL EQUATIONS

15

defined on the interval |x − x0 | < h where h = min(a, b/M ),

M = max |f (x, y)|, (x, y) ∈ R.

Note that this theorem indicates that a solution may not be defined for all x in the interval |x − x0 | ≤ a. For example, the function y=

bCeabx 1 + Ceabx

is solution to y 0 = ay(b − y) but not defined when 1 + Ceabx = 0 even though f (x, y) = ay(b − y satisfies the conditions of the theorem for all x, y. The next example show why the condition on the partial derivative in the above theorem is necessary. Consider the differential equation y 0 = y 1/3 . Again y = 0 is a solution. Separating variables and integrating, we get Z

dy = x + C1 y 1/3

which yields y 2/3 = 2x/3 + C and hence y = ±(2x/3 + C)3/2 . Taking C = 0, we get the solution y = (2x/3)3/2 , (x ≥ 0) which along with the solution y = 0 satisfies y(0) = 0. So the initial value problem y 0 = y 1/3 , y(0) = 0 does not have a unique solution. The reason this 2/3 is not continuous when is so is due to the fact that ∂f ∂y (x, y) = 1/3y y = 0. Many differential equations become linear or separable after a change of variable. We now give two examples of this.

5.

Bernoulli Equation: y 0 = p(x)y + q(x)y n

(n 6= 1).

Note that y = 0 is a solution. To solve this equation, we set u = y α , where α is to be determined. Then, we have u0 = αy α−1 y 0 , hence, our differential equation becomes u0 /α = p(x)u + q(x)y α+n−1 .

(2.5)

Now set α = 1 − n. Thus, (2.5) is reduced to u0 /α = p(x)u + q(x),

(2.6)

which is linear. We know how to solve this for u from which we get solve u = y 1−n to get y.

16

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

6.

Homogeneous Equation: y 0 = F (y/x).

To solve this we let u = y/x so that y = xu and y 0 = u+xu0 . Substituting for y, y 0 in our DE gives u + xu0 = F (u) which is a separable equation. Solving this for u gives y via y = xu. Note that u = a is a solution of xu0 = F (u) − u whenever F (a) = a and that this gives y = ax as a solution of y 0 = f (y/x). Example. y 0 = (x − y)/x + y. This is a homogeneous equation since x−y 1 − y/x = . x+y 1 + y/x Setting u = y/x, our DE becomes xu0 + u =

1−u 1+u

so that

1 − 2u − u2 1−u −u= . 1+u 1+u √ Note that the right-hand side is zero if u = −1± 2. Separating variables and integrating with respect to x, we get xu0 =

Z

(1 + u)du = ln |x| + C1 1 − 2u − u2

which in turn gives (−1/2) ln |1 − 2u − u2 | = ln |x| + C1 . Exponentiating, we get 1 = eC1 |x|. |1 − 2u − u2 |

p

Squaring both sides and taking reciprocals, we get u2 + 2u − 1 = C/x2 with C = ±1/e2C1 . This equation can be solved for u using the quadratic formula. If x0 , y0 are given with x0 6= 0 and u0 = y0 /x0 6= −1 there is, by the fundamental, existence and uniqueness theorem,a unique solution with u(x0 ) = y0 . For example, if x0 = 1, y0 = 2, we have C = 7 and hence u2 + 2u − 1 = 7/x2

17

FIRST ORDER DIFFERENTIAL EQUATIONS

Solving for u, we get

q

u = −1 +

2 + 7/x2

where the positive sign in the quadratic formula was chosen to make u = 2, x = 1 a solution. Hence q

y = −x + x 2 + 7/x2 = −x +

p

2x2 + 7

is the solution to the initial value problem y0 =

x−y , x+y

y(1) = 2

for x > 0 and one can easily check that it is a solution for all x. Moreover, using the fundamental uniqueness, it can be shown that it is the only solution defined for all x.

PART (III): EXACT EQUATION AND INTEGRATING FACTOR — NONLINEAR EQUATIONS (2)

7.

Exact Equations.

By a region of the xy-plane we mean a connected open subset of the plane. The differential equation M (x, y) + N (x, y)

dy =0 dx

is said to be exact on a region (R) if there is a function F (x, y) defined on (R) such that ∂F = M (x, y); ∂x

∂F = N (x, y) ∂y

In this case, if M, N are continuously differentiable on (R) we have ∂M ∂N = . ∂y ∂x

(2.7)

Conversely, it can be shown that condition (2.7) is also sufficient for the exactness of the given DE on (R) providing that (R) is simply connected, .i.e., has no “holes”. The exact equations are solvable. In fact, suppose y(x) is its solution. Then one can write: M [x, y(x)] + N [x, y(x)]

∂F ∂F dy d dy = + = F [x, y(x)] = 0. dx ∂x ∂y dx dx

It follows that F [x, y(x)] = C,

19

20

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

where C is an arbitrary constant. This is an implicit form of the solution y(x). Hence, the function F (x, y), if it is found, will give a family of the solutions of the given DE. The curves F (x, y) = C are called integral curves of the given DE. dy Example 1. 2x2 y dx + 2xy 2 + 1 = 0. Here M = 2xy 2 + 1, N = 2x2 y and R = R2 , the whole xy-plane. The equation is exact on R2 since R2 is simply connected and

∂M ∂N = 4xy = . ∂y ∂x To find F we have to solve the partial differential equations ∂F = 2xy 2 + 1, ∂x

∂F = 2x2 y. ∂y

If we integrate the first equation with respect to x holding y fixed, we get F (x, y) = x2 y 2 + x + φ(y). Differentiating this equation with respect to y gives ∂F = 2x2 y + φ0 (y) = 2x2 y ∂y using the second equation. Hence φ0 (y) = 0 and φ(y) is a constant function. The solutions of our DE in implicit form is x2 y 2 + x = C. Example 2. We have already solved the homogeneous DE dy x−y = . dx x+y This equation can be written in the form y − x + (x + y)

dy =0 dx

which is an exact equation. In this case, the solution in implicit form is x(y − x) + y(x + y) = C, i.e., y 2 + 2xy − x2 = C.

8.

Theorem. If F (x, y) is homogeneous of degree n then x

∂F ∂F +y = nF (x, y). ∂x ∂y

21

FIRST ORDER DIFFERENTIAL EQUATIONS

Proof. The function F is homogeneous of degree n if F (tx, ty) = tn F (x, y). Differentiating this with respect to t and setting t = 1 yields the result. QED

9.

Integrating Factors.

If the differential equation M + N y 0 = 0 is not exact it can sometimes be made exact by multiplying it by a continuously differentiable function µ(x, y). Such a function is called an integrating factor. An integrating ∂µN factor µ satisfies the PDE ∂µM ∂y = ∂x which can be written in the form µ

∂M ∂N − ∂y ∂x

¶

µ=N

∂µ ∂µ −M . ∂x ∂y

This equation can be simplified in special cases, two of which we treat next. µ is a function of x only.

This happens if and only if ∂M ∂y

−

∂N ∂x

N

= p(x)

is a function of x only in which case µ0 = p(x)µ. µ is a function of y only.

This happens if and only if ∂M ∂y

−

∂N ∂x

M

= q(y)

is a function of y only in which case µ0 = −q(y)µ. µ = P (x)Q(y) .

This happens if and only if ∂M ∂N − = p(x)N − q(y)M, ∂y ∂x

where p(x) =

P 0 (x) , P (x)

q(y) =

(2.8)

Q0 (y) . Q(y)

If the system really permits the functions p(x), q(y), such that (2.8) hold, then we can derive R

P (x) = ±e

p(x)dx

R

;

Q(y) = ±e

q(y)dy

.

22

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

Example 1. 2x2 + y + (x2 y − x)y 0 = 0. Here ∂M ∂y

−

∂N ∂x

N

=

2 − 2xy −2 = x2 y − x x

so that there is an integrating factor µ which is a function of x only which satisfies µ0 = −2µ/x. Hence µ = 1/x2 is an integrating factor and 2 + y/x2 + (y − 1/x)y 0 = 0 is an exact equation whose general solution is 2x − y/x + y 2 /2 = C or 2x2 − y + xy 2 /2 = Cx. Example 2. y + (2x − yey )y 0 = 0. Here ∂M ∂y

− M

∂N ∂x

=

−1 y

so that there is an integrating factor which is a function of y only which satisfies µ0 = 1/y. Hence y is an integrating factor and y 2 + (2xy − y 2 ey )y 0 = 0 is an exact equation with general solution xy 2 + (−y 2 + 2y − 2)ey = C. A word of caution is in order here. The solutions of the exact DE obtained by multiplying by the integrating factor may have solutions which are not solutions of the original DE. This is due to the fact that µ may be zero and one will have to possibly exclude those solutions where µ vanishes. However, this is not the case for the above Example 2.

PART (IV): CHANGE OF VARIABLES — NONLINEAR EQUATIONS (3)

10.

Change of Variables.

Sometimes it is possible by means of a change of variable to transform a DE into one of the known types. For example, homogeneous equations can be transformed into separable equations and Bernoulli equations can be transformed into linear equations. The same idea can be applied to some other types of equations, as described as follows.

10.1

y 0 = f (ax + by), b 6= 0

Here, if we make the substitution u = ax+by the differential equation becomes du = bf (u) + a dx which is separable. √ √ Example 1. The DE y 0 = 1 + y − x becomes u0 = u after the change of variable u = y − x.

10.2

dy dx

=

a1 x + b1 y + c1 a2 x + b2 y + c2

Here, we assume that a1 x+b1 y +c1 = 0, a2 x+b2 y +c2 = 0 are distinct lines meeting in the point (x0 , y0 ). The above DE can be written in the form a1 (x − x0 ) + b1 (y − y0 ) dy = dx a2 (x − x0 ) + b2 (y − y0 ) which yields the DE a1 X + b1 Y dY = dX a2 X + b2 Y

23

24

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

after the change of variables X = x − x0 , Y = y − y0 .

10.3

Riccatti equation: y 0 = p(x)y + q(x)y 2 + r(x)

Suppose that u = u(x) is a solution of this DE and make the change of variables y = u + 1/v. Then y 0 = u0 − v 0 /v 2 and the DE becomes u0 − v 0 /v 2 = p(x)(u + 1/v) + q(x)(u2 + 2u/v + 1/v 2 ) + r(x) = p(x)u + q(x)u2 + r(x) + (p(x) + 2uq(x))/v + q(x)/v 2 from which we get v 0 + (p(x) + 2uq(x))v = −q(x), a linear equation. Example 2. y 0 = 1 + x2 − y 2 has the solution y = x and the change of variable y = x + 1/v transforms the equation into v 0 + 2xv = 1.

PART (V): SOME APPLICATIONS

We now give a few applications of differential equations.

11.

Orthogonal Trajectories.

An important application of first order DE’s is to the computation of the orthogonal trajectories of a family of curves f (x, y, C) = 0. An orthogonal trajectory of this family is a curve that, at each point of intersection with a member of the given family, intersects that member orthogonally. To find the orthogonal trajectories, we may derive the ODE, whose solutions are described by these trajectories. For this purpose, we are going first to derive the ODE, whose solutions have the implicit form, f (x, y, C) = 0. In doing so, we differentiate f (x, y, C) = 0 implicitly with respect to x we get ∂f ∂f 0 + y =0 ∂x ∂y from which we get y0 = −

fx (x, y, C) . fy (x, y, C)

Now we solve for C = C(x, y) from the equation f (x, y, C) = 0, which specifies the curve passing through the point (x, y). We substitute C(x, y) in the above formula for y 0 . This gives the equation: h

y 0 = g(x, y) = −

i

fx x, y, C(x, y) h

i.

fy x, y, C(x, y)

Note that y 0 (x) yields the slope of the tangent line at the point (x, y) of a curve of the given family passing through (x, y). The slope of the

25

26

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

orthogonal trajectory at the passing point (x, y) must be y 0 (x) = −

1 . g(x, y)

Therefore, the ODE governing the orthogonal trajectories is derived as h

y0 =

i

fy x, y, C(x, y) h

i.

fx x, y, C(x, y)

Example 3. Let us find the orthogonal trajectories of the family x2 + y 2 = Cx, the family of circles with center on the x-axis and passing through the origin. Here x2 + y 2 x 0 from which, we derive the ODE: y = g(x, y) = (y 2 − x2 )/2xy. Then the ODE governing the orthogonal trajectories can be written as 1 y0 = − , g(x, y) or, y 0 = 2xy/(x2 − y 2 ). 2x + 2yy 0 = C =

The above can be re-written in the form: 2xy + (y 2 − x2 )y 0 = 0. If we let M = 2xy, N = y 2 − x2 we have ∂M ∂y

−

∂N ∂x

M

=

4x 2 = 2xy y

so that we have an integrating factor µ which is a function of y. We have µ0 = −2µ/y from which µ = 1/y 2 . Multiplying the DE for the orthogonal trajectories by 1/y 2 we get Ã

2x x2 + 1− 2 y y

!

y 0 = 0.

∂F 2 2 2 Solving ∂F ∂x = 2x/y, ∂y = 1 − x /y for F yields F (x, y) = x /y + y from which the orthogonal trajectories are x2 /y + y = C, i.e., x2 + y 2 = Cy. This is the family of circles with center on the y-axis and passing through the origin. Note that the line y = 0 is also an orthogonal trajectory that was not found by the above procedure. This is due to the fact that the integrating factor was 1/y 2 which is not defined if y = 0 so we had to work in a region which does not cut the x-axis, e.g., y > 0 or y < 0.

FIRST ORDER DIFFERENTIAL EQUATIONS

12.

27

Falling Bodies with Air Resistance

Let x be the height at time t of a body of mass m falling under the influence of gravity. If g is the force of gravity and b v is the force on the body due to air resistance, Newton’s Second Law of Motion gives the DE dv m = mg − bv dt where v = dx dt . This DE has the general solution v(t) =

mg + Be−bt/m . b

The limit of v(t) as t → ∞ is mg/b, the terminal velocity of the falling body. Integrating once more, we get x(t) = A +

13.

mg t mB −bt/m − e . b b

Mixing Problems

Suppose that a tank is being filled with brine at the rate of a units of volume per second and at the same time b units of volume per second are pumped out. If the concentration of the brine coming in is c units of weight per unit of volume. If at time t = t0 the volume of brine in the tank is V0 and contains x0 units of weight of salt, what is the quantity of salt in the tank at any time t, assuming that the tank is well mixed? If x is the quantity of salt at any time t, we have ac units of weight of salt coming in per second and b

bx x(t) = V (t) V0 + (a − b)(t − t0 )

units of weight of salt going out. Hence dx bx = ac − , dt V0 + (a − b)(t − t0 ) a linear equation. If a = b it has the solution x(t) = cV0 + (x0 − cV0 )e−a(t−t0 )/V0 . As a numerical example, suppose a = b = 1 liter/min, c = 1 grams/liter, V0 = 1000 liters, x0 = 0 and t0 = 0. Then x(t) = 1000(1 − e−.001t ) is the quantity of salt in the tank at any time t. Suppose that after 100 minutes the tank springs a leak letting out an additional liter of brine

28

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

per minute. To find out how much salt is in the tank 12 hours after the leak begins we use the DE dx 2x 2 =1− =1− x. dt 1000 − (t − 100) 1100 − t This equation has the general solution x(t) = (1100 − t)−1 + C(1100 − t)2 . Using x(100) = 1000(1 − e−.1 ) = 95.16, we find C = −9.048 × 10−4 and x(820) = 177.1. When t = 1100 the tank is empty and the differential equation is no a valid description of the physical process. The concentration at time 100 < t < 1100 is x(t) = 1 + C(1100 − t) 1100 − t which converges to 1 as t tends to 1100.

14.

Heating and Cooling Problems

Newton’s Law of Cooling states that the rate of change of the temperature of a cooling body is proportional to the difference between its temperature T and the temperature of its surrounding medium. Assuming the surroundings maintain a constant temperature Ts , we obtain the differential equation dT = −k(T − Ts ), dt where k is a constant. This is a linear DE with solution T = Ts + Ce−kt . If T (0) = T0 then C = T0 − Ts and T = Ts + (T0 − Ts )e−kt . As an example consider the problem of determining the time of death of a healthy person who died in his home some time before noon when his body was 70 degrees. If his body cooled another 5 degrees in 2 hours when did he die, assuming that the room was a constant 60 degrees. Taking noon as t = 0 we have T0 = 70. Since Ts = 60, we get 65 − 60 = 10e−2k from which k = ln(2)/2. To determine the time of death we use the equation 98.6 − 60 = 10e−kt which gives t = − ln(3.86)/k = −2 ln(3.86)/ ln(2) = −3.90. Hence the time of death was 8 : 06 AM.

FIRST ORDER DIFFERENTIAL EQUATIONS

15.

29

Radioactive Decay

A radioactive substance decays at a rate proportional to the amount of substance present. If x is the amount at time t we have dx = −kx, dt where k is a constant. The solution of the DE is x = x(0)e−kt . If c is the half-life of the substance we have by definition x(0)/2 = x(0)e−kc which gives k = ln(2)/c.

PART (VI)*: GEOMETRICAL APPROACHES — NONLINEAR EQUATIONS (4)

16. 16.1

Definitions and Basic Concepts Directional Field

A plot of short line segments drawn at various points in the (x, y) plane showing the slope of the solution curve there is called direction field for the DE.

16.2

Integral Curves

The family of curves in the (x, y) plane, that represent all solutions of DE is called the integral curves.

16.3

Autonomous Systems

The first order DE dy/dx = f (y) is called autonomous, since the independent variable does not appear explicitly. The isoclines are made up of horizonal lines y = m, along which the slope of directional fields is the constant, y 0 = f (m).

16.4

Equilibrium Points

The DE has the constant solution y = y0 , if and only if f (y0 ) = 0. These values of y0 are the equilibrium points or stationary points of the DE. y = y0 is called a source if f (y) changes sign from - to + as y increases from just below y = y0 to just above y = y0 and is called a sink if f (y) changes sign from + to - as y increases from just below y = y0 to just above y = y0 ; it is called a node if there is no change in

31

32

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

sign. Solutions y(t) of the DE appear to be attracted by the line y = y0 , if y0 is a sink and move away from the line y = y0 , if y0 is a source.

17.

Phase Line Analysis

The y-axis on which is plotted the equilibrium points of the DE with arrows between these points to indicate when the solution y is increasing or decreasing is called the phase line of the DE. The autonomous DE dy/dx = 2y − y 2 has 0 and 1 as equilibrium points. The point y = 0 is a source and y = 2 is a sink (see Fig.2.1). This DE is a logistic model for a population having 2 as the size of a stable population. The equation dy/dx = −y(2 − y)(3 − y) has three equilibrium states: y = 0, 2, 3. Among them, y = 0, 3 are the sink, while y = 2 is the source (see Fig.2.2). The equation dy/dx = −y(2 − y)2 has two equilibrium states: y = 0, 2. The point y = 0 is a sink, while y = 2 is a node (see Fig.2.3). The sink is stable, source is unstable, whereas the node is semi-stable. The node point of the equation y = f (y) can either disappear, or split into one sink and one source, when the equation is perturbed with a small amount ε and becomes: y = f (y) + ε.

18.

Bifurcation Diagram

Some dynamical system contains a parameter Λ, such as y 0 = f (y, Λ). Then the characteristics of its equilibrium states, such as their number and nature, depends on the value of Λ. Some times, through a special value of Λ = Λ∗ , these characteristics of equilibrium states may change. This Λ = Λ∗ is called the bifurcation point. Example 1. For the logistic population growth model, if the population is reduced at a constant rate s > 0, the DE becomes dy/dx = 2y − y 2 − s which has a source at the larger of the two roots of the equation y 2 − 2y + s = 0

33

FIRST ORDER DIFFERENTIAL EQUATIONS

y 0 = f (y)

y=0

y=2 y

Figure 2.1. Sketch of the phase line for the equation dy/dx = 2y − y 2 , in which y = 0 is a source, y = 2 is a sink.

y 0 = f (y)

y=0

y=2

y=3

y

Figure 2.2. Sketch of the phase line for the equation dy/dx = −y(2 − y)(3 − y), in which y = 0, 3 is a sink, y = 2 is a source.

for s < 2. If s > 2 there is no equilibrium point and the population dies out as y is always decreasing. The point s=2 is called a bifurcation point of the DE. Example 2. Chemical Reaction Model. One has the DE ·

dy/dx = −ay y 2 − where

¸

R − Rc , a

34

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

y 0 = f (y)

y=0

y=2 y

Figure 2.3. Sketch of the phase line for the equation dy/dx = −y(2 − y)2 , in which the point y = 0 is a sink, while y = 2 is a node.

y is the concentration of species A; R is the concentration of some chemical element, and (a, Rc ) are constants (fixed). It is derived that If R < Rc , the system has one equilibrium state y = 0, which is stable; If R > Rc , the system has q three equilibrium states: y = 0, which is c now unstable, and y = ± R−R a , which are stable. For this system, R = Rc is the bifurcation point.

35

FIRST ORDER DIFFERENTIAL EQUATIONS

y 0 = f (y, s)

s = 1.5 y y0 (s) s = 1.0 s = 1.5 s = 2

s

Figure 2.4. Sketch of the bifurcation diagram of the equation dy/dx = y(2 − y) − s, in which the point s = 2 is the bifurcation point.

36

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

y 0 = f (y, R)

y y0 (R)

R < Rc R = Rc

R > Rc

R

Figure of the bifurcation diagram of the equation dy/dx ¤ £ 2.5. Sketch c , in which the point R = Rc is the bifurcation point. −ay y 2 − R−R a

=

PART (VII): NUMERICAL APPROACH AND APPROXIMATIONS

19.

Euler’s Method

In this section we discuss methods for obtaining a numerical solution of the initial value problem y 0 = f (x, y),

y(x0 ) = y0

at equally spaced points x0 , x1 , x2 , . . . , xN = p, . . . where xn+1 − xn = h > 0 is called the step size. In general, the smaller the value of the better the approximations will be but the number of steps required will be larger. We begin by integrating y 0 = f (x, y) between xn and xn+1 . If y(x) = φ(x), this gives φ(xn+1 ) = φ(xn ) +

Z xn+1 xn

f (t, φ(t))dt.

As a first estimate of the integrand we use the value of f (t, φ(t)) at the lower limit xn , namely f (xn , φ(xn )). Now, assuming that we have already found an estimate yn for φ(xn ), we get the estimate yn+1 = yn + hf (xn , yn ) for φ(xn+1 ). It can be shown that |yn − φ(xn )| ≤ Ch, where C is a constant which depends on p.

37

38

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

20.

Improved Euler’s Method

The Euler method can be improved if we use the trapezoidal rule for estimating the above integral. Namely, Z b a

1 F (x)dx = (F (a) + F (b))(b − a). 2

This leads to the estimate yn+1 = yn +

h (f (xn , yn ) + f (xn+1 , yn+1 )). 2

If we now use the Euler approximation yn+1 to compute f (xn+1 , yn+1 ), we get yn+1 = yn +

h (f (xn , yn ) + f (xn + h, yn + hf (xn , yn )). 2

This is known as the improved Euler method. It can be shown that |yn − φ(xn )| ≤ Ch2 . In general, if yn is an approximation for φ(xn ) such that |yn − φ(xn )| ≤ Chp , we say that the approximation is of order p. Thus the Euler method is first order and the improved Euler is second order.

21.

Higher Order Methods

On can obtain higher order approximations by using better approximations for F (t) = f (t, φ(t)) on the interval [xn , xn+1 ]. For example, the Taylor series approximation F (t) = F (xn )+F 0 (xn )(t−xn )+F 00 (xn )(t−xn )2 /2+· · ·+F (p−1) (xn )(t−xn )p−1 /(p−1)! yields the approximation yn+1 = yn + hf1 (xx , yn ) +

h2 hp f2 (xn , yn ) + · · · + fp−1 (xn , yn ), 2 p!

where ·

fk (xn , yn ) = F

(k−1)

∂ ∂ (xn ) = + f (x, y) ∂x ∂y

¸(k−1)

f (xn , yn ).

It can be show that this approximation is of order p. However it is computationally intensive as one has to compute higher derivatives.

39

FIRST ORDER DIFFERENTIAL EQUATIONS

In the case p = 2 this formula was simplified by Runge and Kutta to give the second order midpoint approximation ·

¸

yn+1 = yn + hf xn +

h h , yn + f (xn , yn ) . 2 2

In the case p = 4 they obtained the 4-th order approximation 1 yn+1 = yn + (k1 + 2k2 + 2k3 + k4 ), 6 where k1 k2 k3 k4

= hf (xn , yn ), = hf (xn + h2 , yn + k21 ), = hf (xn + h2 , yn + k22 ), = hf (xn + h, yn + k3 ).

(2.9)

Computationally, it is much simpler than the 4-th order Taylor series approximation from which it is derived.

4() Picard Iteration We assume that f (x, y) and

∂f ∂y

are continuous on the rectangle

R : |x − x0 | ≤ a, |y − y0 | ≤ b Then |f (x, y)| ≤ M , | ∂f ∂y (x, y)| ≤ L on R. The initial value problem 0 y = f (x, y), y(x0 ) = y0 is equivalent to the integral equation y = y0 +

Z x x0

f (t, y(t))dt.

Let the righthand side of the above equation be denoted by T (y). Then our problem is to find a solution to y = T (y) which is a fixed point problem. To solve this problem we take as an initial approximation to y the constant function y0 (x) = y0 and consider the iterations yn = T n (y0 ). The function yn is called the n-th Picard iteration of y0 . For example, for the initial value problem y 0 = x + y 2 , y(0) = 1 we have y1 (x) = 1 + y2 (x) = 1+

Z x 0

Z x 0

(t + 1)dt = 1 + x + x2 /2

(t+(1+t+t2 /2)2 )dt = 1+x+3x2 /2+2x3 /3+x4 /4+x5 /20.

Contrary to the power series approximations we can determine just how good the Picard iterations approximate y. In fact, we will see that the Picard iterations converge to a solution of our initial value problem. More precisely we have the following result:

40

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

4.1 Theorem of Existence and Uniqueness of Solution for IVP The Picard iterations yn = T n (y0 ) converge to a solution y of y 0 = f (x, y), y(x0 ) = y0 on the interval |x−x0 | ≤ h = min(a, b/M ). Moreover |y(x) − yn (x)| ≤ (M/L)ehL (Lh)n+1 /(n + 1)! for |x − x0 | ≤ h and the solution y is unique on this interval. Proof. We have |y1 − y0 | = |

Z x x0

f (t, y0 )| ≤ M |x − x0 |

since |f (x, y)| ≤ M on R. Now |y1 − y0 | ≤ b if |x − x0 | ≤ h. So (x, y1 (x)) is in R if |x−x0 | ≤ h. Similarly, one can show inductively that (x, yn (x)) is in R if |x − x0 | ≤ h. Using the fact that, by the mean value theorem for derivatives, |f (x, z) − f (x, w| ≤ L|z − w| for all (x, w), (x, z) in R, we obtain Z x

|y2 − y1 | = |

x0

Z x

|y3 − y2 | = |

x0

(f (t, y1 ) − f (t, y0 )| ≤ M L|x − x0 |2 /2,

(f (t, y2 ) − f (t, y1 )| ≤ M L2 |x − x0 |3 /6

and by induction |yn − yn−1 | ≤ M Ln−1 |x − x0 |n /n!. Since the series − yn−1 | is bounded above term by term by the convergent series 1 |ynP n (M/L) ∞ 1 (L|x − x0 |) /n!, its n-th partial sum yn − y0 converges, which gives the convergence of yn to a function y. Now since P∞

y = y0 + (y1 − y0 ) + · · · + (yn − yn−1 ) +

∞ X

(yi − yi−1 )

i=n+1

we obtain |y − yn | ≤

∞ X

(M/L)(L(|x − x0 |)i /i! ≤ (M/L)

i=n+1

(Lh)n+1 hL e . (n + 1)!

For the uniqueness, suppose T (z) = z with (x, z(x) in R for |x − x0 | ≤ h. Then Z x y(x) − z(x) = (f (t, y(x)) − f (t, z(x))dt. x0

If |y(x) − z(x)| ≤ A for x − x0 | ≤ h we then obtain as above |y(x) − z(x)| ≤ AL|x − x0 |.

FIRST ORDER DIFFERENTIAL EQUATIONS

41

Now using this estimate, repeat the above to get |y(x) − z(x)| ≤ AL2 |x − x0 |2 /2. Using induction we get that |y(x) − z(x)| ≤ ALn |x − x0 |n /n! which converges to zero for all x. Hence y = z.

QED

The key ingredient in the proof is the Lipschitz Condition |f (x, y) − f (x, z)| ≤ L|y − z|. If f (x, y) is continuous for |x − x0 | ≤ a and all y and satisfies the above Lipschitz condition in this strip the above proof gives the existence and uniqueness of the solution to the initial value problem y 0 = f (x, y), y(x0 ) = y0 on the interval |x − x0 | ≤ a.

Chapter 3 N-TH ORDER DIFFERENTIAL EQUATIONS

43

PART (I): THE FUNDAMENTAL EXISTENCE AND UNIQUENESS THEOREM

In this lecture we will state and sketch the proof of the fundamental existence and uniqueness theorem for the n-th order DE y (n) = f (x, y, y 0 , . . . , y (n−1) ). The starting point is to convert this DE into a system of first order DE’. Let y1 = y, y2 = y 0 , . . . y (n−1) = yn . Then the above DE is equivalent to the system dy1 dx dy2 dx

= y2 = y3 .. .

dyn dx

= f (x, y1 , y2 , . . . , yn ).

(3.1)

More generally let us consider the system dy1 dx dy2 dx

= f1 (x, y1 , y2 , . . . , yn ) = f2 (x, y1 , y2 , . . . , yn ) .. .

dyn dx

= fn (x, y1 , y2 , . . . , yn ). n

(3.2) o

If we let Y = (y1 , y2 , . . . , yn ), F (x, Y ) = f1 (x, Y ), f2 (x, Y ), . . . , fn (x, Y ) and

dY dx

dyn 1 dy2 = ( dy dx , dx , . . . , dx ), the system becomes

dY = F (x, Y ). dx

45

46

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

1.

Theorem of Existence and Uniqueness (I)

∂fi are continuous on the n + 1-dimensional If fi (x, y1 , . . . , yn ) and ∂y j box R : |x − x0 | < a, |yi − ci | < b, (1 ≤ i ≤ n)

for 1 ≤ i, j ≤ n with |fi (x, y)| ≤ M and ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ∂fi ¯ ¯ ∂fi ¯ ¯+¯ ¯ + . . . ¯ ∂fi ¯ < L ¯ ¯ ∂y ¯ ¯ ∂y ¯ ¯ ∂y ¯ 1

2

n

on R for all i, the initial value problem dY = F (x, Y ), dx

Y (x0 ) = (c1 , c2 , . . . , cn )

has a unique solution on the interval |x − x0 | ≤ h = min(a, b/M ). The proof is exactly the same as for the proof for n = 1 if we use the following Lemma in place of the mean value theorem.

1.1

Lemma

If f (x1 , x2 , . . . xn ) and its partial derivatives are continuous on an ndimensional box R, then for any a, b ∈ R we have ¯ ¯¶ ¯ µ¯ ¯ ¯ ¯ ∂f ¯ ∂f ¯ ¯ ¯ |f (a) − f (b)| ≤ ¯ (c)¯ + · · · + ¯ (c)¯¯ |a − b| ∂x ∂x 1

n

where c is a point on the line between a and b and |(x1 , . . . , xn )| = max(|x1 |, . . . , |xn |). The lemma is proved by applying the mean value theorem to the function G(t) = f (ta + (1 − t)b). This gives G(1) − G(0) = G0 (c) for some c between 0 and 1. The lemma follows from the fact that G0 (x) =

∂f ∂f (a1 − b1 ) + · · · + (an − bn ). ∂x1 ∂xn

The Picard iterations Yk (x) defined by Y0 (x) = Y0 = (c1 , . . . , cn ), Yk+1 (x) = Y0 +

Z x x0

F (t, Yk (t))dt,

converge to the unique solution Y and |Y (x) − Yk (x)| ≤ (M/L)ehL hk+1 /(k + 1)!.

47

N-TH ORDER DIFFERENTIAL EQUATIONS

If f1 (x, y1 , . . . , y) , is an L such that

∂fi ∂yj

are continuous in the strip |x − x0 | ≤ a and there

|f (x, Y ) − f (x, Z)| ≤ L|Y − Z| then h can be taken to be a and M = max|f (x, Y0 )|. This happens in the important special case fi (x, y1 , . . . , yn ) = ai1 (x)y1 + · · · + ain (x)yn + bi (x). As a corollary of the above theorem we get the following fundamental theorem for n-th order DE’s.

2.

Theorem of Existence and Uniqueness (II) If f (x, y1 , . . . , yn ) and

∂f ∂yj

are continuous on the box

R : |x − x0 | ≤ a, |yi − ci | ≤ b (1 ≤ i ≤ n) and |f (x, y1 , . . . , yn )| ≤ M on R, then the initial value problem y (n) = f (x, y, y 0 , . . . , y (n−1) ),

y i−1 (x0 ) = ci (1 ≤ 1 ≤ n)

has a unique solution on the interval |x − x0 | ≤ h = max(a, b/M ). Another important application is to the n-th order linear DE a0 (x)y (n) + a1 (x)y (n−1) + · · · + an (x)y = b(x). In this case f1 = y2 , f2 = y3 , fn = p1 (x)y1 + · · · pn (x)yn + q(x) where pi (x) = an−i (x)/a0 (x), q(x) = −b(x)/a0 (x).

3.

Theorem of Existence and Uniqueness (III)

If a0 (x), a1 (x), . . . , an (x) are continuous on an interval I and a0 (x) 6= 0 on I then, for any x0 ∈ I, that is not an endpoint of I, and any scalars c1 , c2 , . . . , cn , the initial value problem a0 (x)y (n) +a1 (x)y (n−1) +· · ·+an (x)y = b(x), has a unique solution on the interval I.

y i−1 (x0 ) = ci (1 ≤ 1 ≤ n)

PART (II): BASIC THEORY OF LINEAR EQUATIONS

In this lecture we give an introduction to several methods for solving higher order differential equations. Most of what we say will apply to the linear case as there are relatively few non-numerical methods for solving nonlinear equations. There are two important cases however where the DE can be reduced to one of lower degree.

3.1

Case (I)

DE has the form: y (n) = f (x, y 0 , y 00 , . . . , y (n−1) ) where on the right-hand side the variable y does not appear. In this case, setting z = y 0 leads to the DE z (n−1) = f (x, z, z 0 , . . . , z (n−2) ) which is of degree n − 1. If this can be solved then one obtains y by integration with respect to x. For example, consider the DE y 00 = (y 0 )2 . Then, setting z = y 0 , we get the DE z 0 = z 2 which is a separable first order equation for z. Solving it we get z = −1/(x + C) or z = 0 from which y = − log(x + C) + D or y = C. The reader will easily verify that there is exactly one of these solutions which satisfies the initial condition y(x0 ) = y0 , y 0 (x0 ) = y00 for any choice of x0 , y0 , y00 which confirms that it is the general solution since the fundamental theorem guarantees a unique solution.

49

50

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

3.2

Case (II)

DE has the form: y (n) = f (y, y 0 , y 00 , . . . , y (n−1) ) where the independent variable x does not appear explicitly on the righthand side of the equation. Here we again set z = y 0 but try for a solution d d z as a function of y. Then, using the fact that dx = z dy , we get the DE µ

z

d dy

¶n−1

µ

¶

(z) = f y, z, z

dz d , . . . , (z )n (z) dy dy

which is of degree n − 1. For example, the DE y 00 = (y 0 )2 is of this type and we get the DE dz z = z2 dy which has the solution z = Cey . Hence y 0 = Cey from which −e−y = Cx + D. This gives y = − log(−Cx − D) as the general solution which is in agreement with what we did previously.

4. 4.1

Linear Equations Basic Concepts and General Properties

Let us now go to linear equations. The general form is L(y) = a0 (x)y (n) + a1 (x)y (n−1) + · · · + an (x)y = b(x). The function L is called a differential operator. The characteristic feature of L is that L(a1 y1 + a2 y2 ) = a1 L(y1 ) + a2 L(y2 ). Such a function L is what we call a linear operator. Moreover, if L1 (y) = a0 (x)y (n) + a1 (x)y (n−1) + · · · + an (x)y L2 (y) = b0 (x)y (n) + b1 (x)y (n−1) + · · · + bn (x)y and p1 (x), p2 (x) are functions of x the function p1 L1 + p2 L2 defined by (p1 L1 + p2 L2 )(y) = p1 (x)L1 (y) + p2 (x)L2 (y) (3.3) = [a0 (x) + p2 (x)b0 (x)] y (n) + · · · [p1 (x)an (x) + p2 (x)bn (x)] y is again a linear differential operator. An important property of linear operators in general is the distributive law: L(L1 + L2 ) = LL1 + LL2 ,

(L1 + L2 )L = L1 L + L2 L.

51

N-TH ORDER DIFFERENTIAL EQUATIONS

The linearity of equation implies that for any two solutions y1 , y2 the difference y1 − y2 is a solution of the associated homogeneous equation L(y) = 0. Moreover, it implies that any linear combination a1 y1 + a2 y2 of solutions y1 , y2 of L(y) = 0 is again a solution of L(y) = 0. The solution space of L(y) = 0 is also called the kernel of L and is denoted by ker(L). It is a subspace of the vector space of real valued functions on some interval I. If yp is a particular solution of L(y) = b(x), the general solution of L(y) = b(x) is ker(L) + yp = {y + yp | L(y) = 0}. The differential operator L(y) = y 0 may be denoted by D. The operator L(y) = y 00 is nothing but D2 = D ◦D where ◦ denotes composition of functions. More generally, the operator L(y) = y (n) is Dn . The identity operator I is defined by I(y) = y. By definition D0 = I. The general linear n-th order ODE can therefore be written h

i

a0 (x)Dn + a1 (x)Dn−1 + · · · + an (x)I (y) = b(x).

5.

Basic Theory of Linear Differential Equations

In this lecture we will develop the theory of linear differential equations. The starting point is the fundamental existence theorem for the general n-th order ODE L(y) = b(x), where L(y) = Dn + a1 (x)Dn−1 + · · · + an (x). We will also assume that a0 (x), a1 (x), . . . , an (x), b(x) are continuous functions on the interval I.

5.1

Basics of Linear Vector Space

5.1.1 Isomorphic Linear Transformation From the fundamental theorem, it is known that for any x0 ∈ I, the initial value problem L(y) = b(x)

y(x0 ) = d1 , y 0 (x0 ) = d2 , . . . , y (n−1) (x0 ) = dn

has a unique solution for any d1 , d2 , . . . , dn ∈ R. Thus, if V is the solution space of the associated homogeneous DE L(y) = 0, the transformation T : V → Rn , defined by T (y) = (y(x0 ), y 0 (x0 ), . . . , y (n−1) (x0 )), is linear transformation of the vector space V into Rn since T (ay + bz) = aT (y) + bT (z).

52

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

Moreover, the fundamental theorem says that T is one-to-one (T (y) = T (z) impliesy = z) and onto (every d ∈ Rn is of the form T (y) for some y ∈ V ). A linear transformation which is one-to-one and onto is called an isomorphism. Isomorphic vector spaces have the same properties.

5.1.2 Dimension and Basis of Vector Space We call the vector space being n-dimensional with the notation by dim(V ) = n. This means that there exists a sequence of elements: y1 , y2 , . . . , yn ∈ V such that every y ∈ V can be uniquely written in the form y = c1 y1 + c2 y2 + . . . cn yn with c1 , c2 , . . . , cn ∈ R. Such a sequence of elements of a vector space V is called a basis for V . In the context of DE’s it is also known as a fundamental set. The number of elements in a basis for V is called the dimension of V and is denoted by dim(V ). If e1 = (1, 0, . . . , 0), e2 = (0, 1, . . . , 0), . . . , en = (0, 0, . . . , 1) is the standard basis of Rn and yi is the unique yi ∈ V with T (yi ) = ei then y1 , y2 , . . . , yn is a basis for V . This follows from the fact that T (c1 y1 + c2 y2 + · · · + cn yn ) = c1 T (y1 ) + c2 T (y2 ) + · · · + cn T (yn ).

5.1.3 (*) Span and Subspace A set of vectors v1 , v2 , · · · , vn in a vector space V is said to span or generate V if every v ∈ V can be written in the form v = c1 v1 + c2 v2 + · · · + cn vn with c1 , c2 , . . . , cn ∈ R. Obviously, not any set of n vectors can span the vector space V . It will be seen that {v1 , v2 , · · · , vn } span the vector space V , if and only if they are linear independent. The set S = span(v1 , v2 , . . . , vn ) = {c1 v1 + c2 v2 + · · · + cn vn | c1 , c2 , . . . , cn ∈ R} consisting of all possible linear combinations of the vectors v1 , v2 , . . . , vn form a subspace of V , which may be also called the span of {v1 , v2 , . . . , vn }. Then V = span(v1 , v2 , . . . , vn ) if and only if v1 , v2 , . . . , vn spans V .

5.1.4 Linear Independency The vectors v1 , v2 , . . . , vn are said to be linearly independent if c1 v1 + c2 v2 + . . . cn vn = 0

53

N-TH ORDER DIFFERENTIAL EQUATIONS

implies that the scalars c1 , c2 , . . . , cn are all zero. A basis can also be characterized as a linearly independent generating set since the uniqueness of representation is equivalent to linear independence. More precisely, c1 v1 + c2 v2 + · · · + cn vn = c01 v1 + c02 v2 + · · · + c0n vn implies

ci = c0i

for all i,

if and only if v1 , v2 , . . . , vn are linearly independent. As an example of a linearly independent set of functions consider cos(x), cos(2x), sin(3x). To prove their linear independence, suppose that c1 , c2 , c3 are scalars such that c1 cos(x) + c2 cos(2x) + c3 sin(3x) = 0 for all x. Then setting x = 0, π/2, π, we get c1 + c2 + c3 = 0, −c2 − c3 = 0, −c1 + c2 =0

(3.4)

from which c1 = c2 = c3 = 0. An example of a linearly dependent set would be sin2 (x), cos2 (x), cos(2x) since cos(2x) = cos2 (x) − sin2 (x) implies that cos(2x) + sin2 (x) + (−1) cos2 (x) = 0.

5.2

Wronskian of n-functions

Another criterion for linear independence of functions involves the Wronskian.

5.2.1 Definition If y1 , y2 , . . . , yn are n functions which have derivatives up to order n − 1 then the Wronskian of these functions is the determinant ¯ ¯ y1 ¯ ¯ y0 ¯ 1 W = W (y1 , y2 , . . . , yn ) = ¯¯ .. ¯. ¯ (n−1) ¯y 1

¯ ¯ ¯ ¯ ¯ ¯. ¯ ¯ (n−1) (n−1) ¯¯ y2 . . . yn

y2 y20 .. .

. . . yn . . . yn0 .. .

If W (x0 ) 6= 0 for some point x0 , then y1 , y2 , . . . , yn are linearly independent. This follows from the fact that W (x0 ) is the determinant of

54

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

the coefficient matrix of the linear homogeneous system of equations in c1 , c2 , . . . , cn obtained from the dependence relation c1 y1 + c2 y2 + · · · + cn yn = 0 and its first n − 1 derivatives by setting x = x0 . For example, if y1 = cos(x), cos(2x), cos(3x) we have ¯ ¯ ¯ ¯ cos(x) cos(2x) cos(3x) ¯ ¯ ¯ W = ¯ − sin(x) −2 sin(2x) −3 sin(3x) ¯¯ ¯ − cos(x) −4 cos(2x) −9 cos(3x) ¯

and W (π/4)) = −8 which implies that y1 , y2 , y3 are linearly independent. Note that W (0) = 0 so that you cannot conclude linear dependence from the vanishing of the Wronskian at a point. This is not the case if y1 , y2 , . . . , yn are solutions of an n-th order linear homogeneous ODE.

5.2.2 Theorem 1 The the Wronskian of n solutions of the n-th order linear ODE L(y) = 0 is subject to the following first order ODE: dW = −a1 (x)W, dx with solution W (x) = W (x0 )e

−

Rx x0

a1 (t)dt

.

From the above it follows that the Wronskian of n solutions of the n-th order linear ODE L(y) = 0 is either identically zero or vanishes nowhere.

5.2.3 Theorem 2 If y1 , y2 , . . . , yn are solutions of the linear ODE L(y) = 0, the following are equivalent: 1 y1 , y2 , . . . , yn is a basis for the vector space V = ker(L); 2 y1 , y2 , . . . , yn are linearly independent; 3

(∗)

y1 , y2 , . . . , yn span V ;

4 y1 , y2 , . . . , yn generate ker(L); 5 W (y1 , y2 , . . . , yn ) 6= 0 at some point x0 ; 6 W (y1 , y2 , . . . , yn ) is never zero.

55

N-TH ORDER DIFFERENTIAL EQUATIONS

Proof. The equivalence of 1, 2, 3 follows from the fact that ker(L) is isomorphic to Rn . The rest of the proof follows from the fact that if the Wronskian were zero at some point x0 the homogeneous system of equations c1 y1 (x0 ) + c1 y2 (x0 ) + · · · + cn yn (x0 ) c1 y10 (x0 ) + c1 y20 (x0 ) + · · · + cn yn0 (x0 ) .. . (n−1)

c1 y1

(n−1)

(x0 ) + c1 y2

=0 =0 (n−1)

(x0 ) + · · · + cn yn

(3.5)

(x0 ) = 0

would have a non-zero solution for c1 , c2 , . . . , cn which would imply that c1 y1 + c2 y2 + · · · + cn yn = 0 and hence that y1 , y2 , . . . , yn are not linearly independent.

QED

From the above, we see that to solve the n-th order linear DE L(y) = b(x) we first find linear n independent solutions y1 , y2 , . . . , yn of L(y) = 0. Then, if yP is a particular solution of L(y) = b(x), the general solution of L(y) = b(x) is y = c1 y1 + c2 y2 + · · · + cn yn + yP . (n−1)

The initial conditions y(x0 ) = d1 , y 0 (x0 ) = d2 , . . . , yn determine the constants c1 , c2 , . . . , cn uniquely.

(x0 ) = dn then

PART (III): SOLUTIONS FOR EQUATIONS WITH CONSTANTS COEFFICIENTS (1)

In what follows, we shall first focus on the linear equations with constant coefficients: L(y) = a0 y (n) + a1 y (n−1) + · · · + an y = b(x) and present two different approaches to solve them.

6.

The Method with Undetermined Parameters

To illustrate the idea, as a special case, let us first consider the 2-nd order Linear equation with the constant coefficients: L(y) = ay 00 + by 0 + cy = f (x).

(3.6)

The associate homogeneous equation is: L(y) = ay 00 + by 0 + cy = 0.

6.1

(3.7)

Basic Equalities (I)

We first give the following basic identities: D(erx ) = rerx ; D2 (erx ) = r2 erx ; · · · Dn (erx ) = rn erx .

(3.8)

To solve this equation, we assume that the solution is in the form y(x) = erx , where r is a constant to be determined. Due to the properties of the exponential function erx : y 0 (x) = ry(x); y 00 (x) = r2 y(x); · · · y (n) = rn y(x),

57

58

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

we can write L(erx ) = φ(r)erx .

(3.9)

for any given (r, x), where φ(r) = ar2 + br + c. is called the characteristic polynomial. From (3.9) it is seen that the function erx satisfies the equation (3.6), namely L(erx ) = 0, as long as the constant r is the root of the characteristic polynomial, i.e. φ(r) = 0. In general, the polynomial φ(r) has two roots (r1, r2 ): One can write φ(r) = ar2 + br + c = a(r − r1 )(r − r2 ). Accordingly, the equation (3.7) has two solutions: {y1 (x) = er1 x ; y2 (x) = er2 x }. Two cases should be discussed separately.

6.2

Cases (I) ( r1 > r2 )

When b2 − 4ac > 0, the polynomial φ(r) has two distinct real roots (r1 6= r2 ). In this case, the two solutions, y( x); y2 (x) are different. The following linear combination is not only solution, but also the general solution of the equation: y(x) = Ay1 (x) + By2 (x),

(3.10)

where A, B are arbitrary constants. To prove that, we make use of the fundamental theorem which states that if y, z are two solutions such that y(0) = z(0) = y0 and y 0 (0) = z 0 (0 = y00 ) then y = z. Let y be any solution and consider the linear equations in A, B Ay1 (0) + By2 (0) = y(0), Ay10 (0) + By20 (0) = y 0 (0),

(3.11)

A+B = y0 , Ar1 + Br2 = y00 .

(3.12)

or

Due to r1 6= r2 , these conditions leads to the unique solution A, B. With this choice of A, B the solution z = Ay1 + By2 satisfies z(0) = y(0), z 0 (0) = y 0 (0) and hence y = z. Thus, (3.10) contains all possible solutions of the equation, so, it is indeed the general solution.

59

N-TH ORDER DIFFERENTIAL EQUATIONS

6.3

Cases (II) ( r1 = r2 )

When b2 −4ac = 0, the polynomial φ(r) has double root: r1 = r2 = −b 2a . In this case, the solution y1 (x) = y2 (x) = er1 x . Thus, for the general solution, one needs to derive another type of the second solution. For this purpose, one may use the method of reduction of order. Let us look for a solution of the form C(x)er1 x with the undetermined function C(x). By substituting the equation, we derive that ³

´

h

i

L C(x)er1 x = C(x)φ(r1 )er1 x +a C 00 (x)+2r1 C 0 (x) er1 x +bC 0 (x)er1 x = 0. Noting that φ(r1 ) = 0; we get

2ar1 + b = 0,

C 00 (x) = 0

or C(x) = Ax + B, where A, B are arbitrary constants. Thus, we solution: y(x) = (Ax + B)er1 x ,

(3.13)

is a two parameter family of solutions consisting of the linear combinations of the two solutions y1 = er1 x and y2 = xer1 x . It is also the general solution of the equation. The proof is similar to that given for the case (I) based on the fundamental theorem of existence and uniqueness. Let y be any solution and consider the linear equations in A, B Ay1 (0) + By2 (0) = y(0), Ay10 (0) + By20 (0) = y 0 (0),

(3.14)

A = y(0), Ar1 + B = y 0 (0).

(3.15)

or

these conditions leads to the unique solution A = y(0), B = y 0 (0) − r1 y(0). With this choice of A, B the solution z = Ay1 + By2 satisfies z(0) = y(0), z 0 (0) = y 0 (0) and hence y = z. Thus, (3.13) contains all possible solutions of the equation, so, it is indeed the general solution. The approach presented in this subsection is applicable to any higher order equations with constant coefficients. Example 1. Consider the linear DE y 00 + 2y 0 + y = x. Here L(y) = y 00 + 2y 0 + y. A particular solution of the DE L(y) = x is yp = x − 2. The associated homogeneous equation is y 00 + 2y 00 + y = 0.

60

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

The characteristic polynomial φ(r) = r2 + 2r + 1 = (r + 1)2 has double roots r1 = r2 = −1. Thus the general solution of the DE y 00 + 2y 0 + y = x is y = Axe−x + Be−x + x − 2. This equation can be solved quite simply without the use of the fundamental theorem if we make essential use of operators.

6.4

Cases (III) ( r1;2 = λ ± iµ)

When b2 − 4ac < 0, the polynomial φ(r) has two conjugate complex roots r1,2 = λ ± iµ. We have to define the complex number, i2 = −1;

i3 = −i;

i4 = 1;

i5 = i, · · ·

and define and complex function with the Taylor series: eix =

∞ n n X i x n=0

n!

=

∞ X (−1)n x2n n=0

2n!

+i

∞ X (−1)n x2n+1 n=0

(2n + 1)!

= cos x + i sin x.

(3.16)

From the definition, it follows that ex+iy = ex eiy = ex (cos y + i sin y) . and

D(erx ) = rex ,

Dn (erx ) = rn ex

where r is a complex number. So that, the basic equalities are now extended to the case with complex number r. Thus, we have the two complex solutions: y1 (x) = er1 x = eλx (cos µx+i sin µx),

y2 (x) = er2 x = eλx (cos µx−i sin µx)

with a proper combination of these two solutions, one may derive two real solutions: y˜1 (x) = eλx cos µx,

y˜2 (x) = eλx sin µx

and the general solution: y(x) = eλx (A cos µx + B sin µx).

PART (IV): SOLUTIONS FOR EQUATIONS WITH CONSTANTS COEFFICIENTS (2)

We adopt the differential operator D and write the linear equation in the following form: L(y) = (a0 D(n) + a1 D(n−1) + · · · + an )y = P (D)y = b(x).

7. 7.1

The Method with Differential Operator Basic Equalities (II).

We may prove the following basic identity of differential operators: for any scalar a, (D − a) = eax De−ax (D − a)n = eax Dn e−ax

(3.17)

where the factors eax , e−ax are interpreted as linear operators. This identity is just the fact that µ ¶ dy d −ax − ay = eax (e y) . dx dx The formula (3.17) may be extensively used in solving the type of linear equations under discussion. Let write the equation (3.7) with the differential operator in the following form: L(y) = (aD2 + bD + c)y = φ(D)y = 0, where

(3.18)

φ(D) = (aD2 + bD + c) is a polynomial of D. We now re-consider the cases above-discussed with the previous method.

61

62

7.2

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

Cases (I) ( b2 − 4ac > 0)

The polynomial φ(r) have two distinct real roots r1 > r2 . Then, we can factorize the polynomial φ(D) = (D − r1 )(D − r2 ) and re-write the equation as: L(y) = (D − r1 )(D − r2 )y = 0. letting z = (D − r2 )y, in terms the basic equalities, we derive (D − r1 )z = er1 x De−r1 x z = 0, er1 x z = A,

z = Aer1 x .

Furthermore, from (D − r2 )y = er2 x De−r2 x y = z = Aer1 x , we derive and

D(e−r2 x y) = ze−r2 x = Ae(r1 −r2 )x ˜ r1 x + Ber2 x , y = Ae

A where A˜ = (r1 −r , B are arbitrary constants. It is seen that, in general, 2) to solve the equation

L(y) = (D − r1 )(D − r2 ) · · · (D − rn )y = 0, where ri 6= rj , (i 6= j), one can first solve each factor equations (D − ri )yi = 0,

(i = 1, 2, · · · , n)

separately. The general solution can be written in the form: y(x) = y1 (x) + y2 (x) + · · · + yn (x).

7.3

Cases (II) ( b2 − 4ac = 0)

. The polynomial φ(r) have double real roots r1 = r2 . Then, we can factorize the polynomial φ(D) = (D − r1 )2 and re-write the equation as: L(y) = (D − r1 )2 y = 0. In terms the basic equalities, we derive (D − r1 )2 y = er1 x D2 e−r1 x y = 0,

N-TH ORDER DIFFERENTIAL EQUATIONS

hence,

63

D2 (e−r1 x y) = 0.

One can solve

(e−r1 x y) = A + Bx,

or

y = (A + Bx)er1 x . In general, for the equation, L(y) = (D − r1 )n y = 0.

we have the general solution: y = (A1 + A2 x + · · · + An xn−1 )er1 x . So, we may write ker((D − a)n ) = {(a0 + ax + · · · + an−1 xn−1 )eax | a0 , a1 , . . . , an−1 ∈ R}.

7.4

Cases (III) ( b2 − 4ac < 0)

The polynomial φ(r) have two complex conjugate roots r1,2 = λ ± iµ. Then, we can factorize the polynomial φ(D) = (D−λ)2 +µ2 , and re-write the equation as: L(y) = ((D − λ)2 + µ2 )y = 0. Let us consider the special case first: L(z) = (D2 + µ2 )z = 0. From the formulas: D(cos µx) = −µ sin x,

D(sin x) = µ cos x,

it follows that z(x) = A cos µx + B sin µx. To solve for y(x), we re-write the equation (3.19) as (eλx D2 e−λx + µ2 )y = 0. Then

D2 (e−λx y) + µ2 e−λx y = (D2 + µ2 )e−λx y = 0.

Thus, we derive e−λx y(x) = A cos µx + B sin µx,

(3.19)

64

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

or y(x) = eλx (A cos µx + B sin µx).

(3.20)

One may also consider case (I) with the complex number r1 , r2 and obtain the complex solution: y(x) = eλx (Aeiµx + Be−iµx ).

7.5

(3.21)

Theorems

In summary, it can be proved that the following results hold: ker ((D − a)m ) = span(eax , xeax , . . . , xm−1 eax ) It means that ((D − a)m )y = 0 has a set of fundamental solutions: n

eax , xeax , . . . , xm−1 eax

o

ker ((D − a)2 + b2 )m ) = span(eax f (x), xeax f (x), . . . , xm−1 eax f (x)), f (x) = cos(bx) or sin(bx) It means that ((D − a)2 + b2 )m )y = 0 has a set of fundamental solutions: n o eax f (x), xeax f (x), . . . , xm−1 eax f (x) , where f (x) = cos(bx) or sin(bx). ker(P (D)Q(D)) = ker(P (D))+ker(Q(D)) = {y1 +y2 | y1 ∈ ker(P (D)), y2 ∈ ker(Q(D))}, if P (X), Q(X) are two polynomials with constant coefficients that have no common root. It means that if P (X), Q(X) have no common roots, then the set of fundamental solutions for the operator P (D)Q(D) is just the joint set: {p1 (x), p2 (x), · · · , pn (x); q1 (x), q2 (x), · · · qm (x)}, where {p1 (x), p2 (x), · · · , pn (x)} is the set of fundamental solutions for the operator P (D), and {q1 (x), q2 (x), · · · qm (x)} is the set of fundamental solutions for the operator Q(D). Example 1. By using the differential operation method, one can easily solve some inhomogeneous equations. For instance, let us reconsider the example 1. One may write the DE y 00 + 2y 0 + y = x in the operator form as (D2 + 2D + I)(y) = x.

N-TH ORDER DIFFERENTIAL EQUATIONS

65

The operator (D2 + 2D + I) = φ(D) can be factored as (D + I)2 . With (3.17), we derive that (D + I)2 = (e−x Dex )(e−x Dex ) = e−x D2 ex . Consequently, the DE (D + I)2 (y) = x can be written e−x D2 ex (y) = x or d2 x (e y) = xex dx which on integrating twice gives ex y = xex − 2ex + Ax + B,

y = x − 2 + Axe−x + Be−x .

We leave it to the reader to prove that ker((D − a)n ) = {(a0 + ax + · · · + an−1 xn−1 )eax | a0 , a1 , . . . , an−1 ∈ R}. Example 2. Now consider the DE y 00 − 3y 0 + 2y = ex . In operator form this equation is (D2 − 3D + 2I)(y) = ex . Since (D2 − 3D + 2I) = (D − I)(D − 2I), this DE can be written (D − I)(D − 2I)(y) = ex . Now let z = (D − 2I)(y). Then (D − I)(z) = ex , a first order linear DE which has the solution z = xex + Aex . Now z = (D − 2I)(y) is the linear first order DE y 0 − 2y = xex + Aex which has the solution y = ex − xex − Aex + Be2x . Notice that −Aex + Be2x is the general solution of the associated homogeneous DE y 00 − 3y 0 + 2y = 0 and that ex − xex is a particular solution of the original DE y 00 − 3y 0 + 2y = ex . Example 3. Consider the DE y 00 + 2y 0 + 5y = sin(x) which in operator form is (D2 + 2D + 5I)(y) = sin(x). Now D2 + 2D + 5I = (D + I)2 + 4I and so the associated homogeneous DE has the general solution Ae−x cos(2x) + Be−x sin(2x).

66

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

All that remains is to find a particular solution of the original DE. We leave it to the reader to show that there is a particular solution of the form C cos(x) + D sin(x). Example 4. Solve the initial value problem y 000 − 3y 00 + 7y 0 − 5y = 0,

y(0) = 1, y 0 (0) = y 00 (0) = 0.

The DE in operator form is (D3 − 3D2 + 7D − 5)(y) = 0. Since φ(r) = r3 − 3r2 + 7r − 5 = (r − 1)(r2 − 2r + 5) = (r − 1)[(r − 1)2 + 4], we have L(y) = = = =

(D3 − 3D2 + 7D − 5)(y) (D − 1)[(D − 1)2 + 4](y) [(D − 1)2 + 4](D − 1)(y) 0.

(3.22)

From here, it is seen that the solutions for (D − 1)(y) = 0,

(3.23)

y(x) = c1 ex ,

(3.24)

[(D − 1)2 + 4](y) = 0,

(3.25)

y(x) = c2 ex cos(2x) + c3 ex sin(2x),

(3.26)

namely, and the solutions for namely, must be the solutions for our equation (3.22). Thus, we derive that the following linear combination y = c1 ex + c2 ex cos(2x) + c3 ex sin(2x),

(3.27)

must be the solutions for our equation (3.22). In solution (3.27), there are three arbitrary constants (c1 , c2 , c3 ). One can prove that this solution is the general solution, which covers all possible solutions of (3.22). For instance, given the I.C.’s: y(0) = 1, y 0 (0) = 0, y 00 (0) = 0, from (3.27), we can derive c1 + c2 = 1, c1 + c2 + 2c3 = 0, c1 − 3c2 + 4c3 = 0, and find c1 = 5/4, c2 = −1/4, c3 = −1/2.

PART (V): FINDING A PARTICULAR SOLUTION FOR INHOMOGENEOUS EQUATION

Variation of parameters is method for producing a particular solution of a special kind for the general linear DE in normal form L(y) = y (n) + a1 (x)y (n−1) + · · · + an (x)y = b(x) from a fundamental set y1 , y2 , . . . , yn of solutions of the associated homogeneous equation.

8.

The Differential Operator for Equations with Constant Coefficients Given L(y) = P (D)(y) = (a0 D(n) + a1 D(n−1) + · · · + an D)y = b(x).

Assume that the inhomogeneous term b(x) is a solution of linear equation: Q(D)(b(x)) = 0. Then we can transform the original inhomogeneous equation to the homogeneous equation by applying the differential operator Q(D) to its both sides, Φ(D)(y) = Q(D)P (D)(y) = 0. The operator Q(D) is called the Annihilator. The above method is also called the Annihilator Method. Example 1. Solve the initial value problem y 000 − 3y 00 + 7y 0 − 5y = x + ex ,

67

y(0) = 1, y 0 (0) = y 00 (0) = 0.

68

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

This DE is non-homogeneous. The associated homogeneous equation was solved in the previous lecture. Note that in this example, In the inhomogeneous term b(x) = x + ex is in the kernel of Q(D) = D2 (D − 1). Hence, we have D2 (D − 1)2 ((D − 1)2 + 4)(y) = 0 which yields y = Ax+B +Cxex +c1 ex +c2 ex cos(2x)+c3 ex sin(2x). This shows that there is a particular solution of the form yP = Ax+B +Cxex which is obtained by discarding the terms in the solution space of the associated homogeneous DE. Substituting this in the original DE we get y 000 − 3y 00 + 7y 0 − 5y = 7A − 5B − 5Ax − Cex which is equal to x + ex if and only if 7A − 5B = 0, −5A = 1, −C = 1 so that A = −1/5, B = −7/25, C = −1. Hence the general solution is y = c1 ex + c2 ex cos(2x) + c3 ex sin(2x) − x/5 − 7/25 − xex . To satisfy the initial condition y(0) = 0, y 0 (0) = y 00 (0) = 0 we must have c1 + c2 = 32/25, c1 + c2 + 2c3 = 6/5, c1 − 3c2 + 4c3 = 2

(3.28)

which has the solution c1 = 3/2, c2 = −11/50, c3 = −1/25. It is evident that if the function b(x) can not be eliminated by any linear operator Q(D), the annihilator method will not applicable.

9.

The Method of Variation of Parameters In this method we try for a solution of the form yP = C1 (x)y1 + C2 (x)y2 + · · · + Cn (x)yn .

Then yP0 = C1 (x)y10 + C2 (x)y20 + · · · + Cn (x)yn0 + C10 (x)y1 + C20 (x)y2 + · · · + Cn0 (x)yn and we impose the condition C10 (x)y1 + C20 (x)y2 + · · · + Cn0 (x)yn = 0. Then yP0 = C1 (x)y10 + C2 (x)y20 + · · · + Cn (x)yn0 and hence yP00 = C1 (x)y100 +C2 (x)y200 +· · ·+Cn (x)yn00 +C10 (x)y10 +C20 (x)y20 +· · ·+Cn0 (x)yn0 . Again we impose the condition C10 (x)y10 + C20 (x)y20 + · · · + Cn0 (x)yn0 = 0 so that yP00 = C1 (x)y100 + C2 (x)y200 + · · · + Cn (x)yn0 .

69

N-TH ORDER DIFFERENTIAL EQUATIONS

We do this for the first n − 1 derivatives of y so that for 1 ≤ k ≤ n − 1 (k)

(k)

(k)

yP = C1 (x)y1 + C2 (x)y2 + · · · Cn (x)yn(k) , (k)

(k)

C10 (x)y1 + C20 (x)y2 + · · · + Cn0 (x)yn(k) = 0. (n−1)

Now substituting yP , yP0 , . . . , yP

in L(y) = b(x) we get (n−1)

C1 (x)L(y1 ) + C2 (x)L(y2 ) + · · · + Cn (x)L(yn ) + C10 (x)y1 (n−1) +C20 (x)y2

(3.29)

+ · · · + Cn0 (x)yn(n−1) = b(x).

But L(yi ) = 0 for 1 ≤ k ≤ n so that (n−1)

C10 (x)y1

(n−1)

+ C20 (x)y2

+ · · · + Cn0 (x)yn(n−1) = b(x).

We thus obtain the system of n linear equations for C10 (x), . . . , Cn0 (x) C10 (x)y1 + C20 (x)y2 + · · · + Cn0 (x)yn C10 (x)y10 + C20 (x)y20 + · · · + Cn0 (x)yn0 .. . (n−1)

C10 (x)y1

(n−1)

+ C20 (x)y2

= 0, = 0, (n−1)

+ · · · + Cn0 (x)yn

(3.30)

= b(x).

If we solve this system using Cramer’s Rule and integrate, we find Ci (x) =

Z x x0

(−1)n+i b(t)

Wi dt W

where W = W (y1 , y2 , . . . , yn ) and Wi = W (y1 . . . , yˆi , . . . , yn ) where theˆ means that yi is omitted. Note that the particular solution yP found in this way satisfies (n−1)

yP (x0 ) = yP0 (x0 ) = · · · = yP

= 0.

The point x0 is any point in the interval of continuity of the ai (x) and b(x). Note that yP is a linear function of the function b(x). Example 2. Find the general solution of y 00 + y = 1/x on x > 0. The general solution of y 00 + y = 0 is y = c1 cos(x) + c2 sin(x). Using variation of parameters with y1 = cos(x), y2 = sin(x), b(x) = 1/x and x0 = 1, we have W = 1, W1 = sin(x), W2 = cos(x) and we obtain the particular solution yp = C1 (x) cos(x) + C2 (x) sin(x) where C1 (x) = −

Z x sin(t) 1

t

dt,

C2 (x) =

Z x cos(t) 1

t

dt.

70

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

The general solution of y 00 + y = 1/x on x > 0 is therefore y = c1 cos(x) + c2 sin(x) −

µZ x sin(t)

t

1

¶

dt cos(x) +

µZ x cos(t)

t

1

¶

dt sin(x).

When applicable, the annihilator method is easier as one can see from the DE y 00 + y = ex . Here it is immediate that yp = ex /2 is a particular solution while variation of parameters gives yp = −

µZ x 0

¶ t

e sin(t)dt cos(x) +

µZ x 0

¶ t

e cos(t)dt sin(x).

The integrals can be evaluated using integration by parts: Rx t Rx t x 0 e cos(t)dt = e cos(x) − 1 + 0 e sin(t)dt R

= ex cos(x) + ex sin(x) − 1 −

x t 0 e cos(t)dt

(3.31)

which gives Z x 0

Z x 0

h

i

et cos(t)dt = ex cos(x) + ex sin(x) − 1 /2

et sin(t)dt = ex sin(x) −

Z x 0

h

i

et cos(t)dt = ex sin(x) − ex cos(x) + 1 /2

so that after simplification yp = ex /2 − cos(x)/2 − sin(x)/2.

PART (VI): LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS

In this lecture we will give a few techniques for solving certain linear differential equations with non-constant coefficients. We will mainly restrict our attention to second order equations. However, the techniques can be extended to higher order equations. The general second order linear DE is p0 (x)y 00 + p1 (x)y 0 + p2 (x)y = q(x). This equation is called a non-constant coefficient equation if at least one of the functions pi is not a constant function.

10.

Euler Equations

An important example of a non-constant linear DE is Euler’s equation x2 y 00 + axy 0 + by = 0, where a, b are constants. This equation has singularity at x = 0. The fundamental theorem of existence and uniqueness of solution holds in the region x > 0 and x, 0, respectively. So one must solve the problem in the region x > 0, or x < 0 separately. We first consider the region x > 0. This Euler equation can be transformed into a constant coefficient DE by the change of independent variable x = et . This is most easily seen by noting that dy dx dy dy = = et = xy 0 dt dx dt dx so that

dy dx

= e−t dy dt . In operator form, we have d d d = et =x . dt dx dx

71

72

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

If we set D =

d dt ,

we have

d dx

= e−t D so that

d2 = e−t De−t D = e−2t et De−t D = e−2t (D − 1)D dx2 so that x2 y 00 = D(D − 1). By induction one easily proves that dn = e−nt D(D − 1) · · · (D − n + 1) dxn or xn y (n) = D(D − 1) · · · (D − n + 1)(y). With the variable t, Euler’s equation becomes d2 y dy + (a − 1) + by = q(et ), 2 dt dt which is a linear constant coefficient DE. Solving this for y as a function of t and then making the change of variable t = ln(x), we obtain the solution of Euler’s equation for y as a function of x. For the region x < 0, we may let −x = et , or |x| = et . Then the equation x2 y 00 + axy 0 + by = 0, (x < 0) is changed to the same form d2 y dy + (a − 1) + by = 0. dt2 dt Hence, we have the solution y(t) = y(ln |x|) (x < 0). The above approach, can extend to solve the n-th order Euler equation xn y (n) + a1 xn−1 y (n−1) + · · · + an y = q(x), where a1 , a2 , . . . an are constants. Example 1. Solve x2 y 00 + xy 0 + y = ln(x), (x > 0). Making the change of variable x = et we obtain d2 y +y =t dt2 whose general solution is y = A cos(t) + B sin(t) + t. Hence y = A cos(ln(x)) + B sin(ln(x)) + ln(x) is the general solution of the given DE.

73

N-TH ORDER DIFFERENTIAL EQUATIONS

Example 2. Solve x3 y 000 + 2x2 y 00 + xy 0 − y = 0,

(x > 0).

This is a third order Euler equation. Making the change of variable x = et , we get ³

´

D(D − 1)(D − 2) + 2D(D − 1) + (D − 1) (y) = (D − 1)(D2 + 1)(y) = 0

which has the general solution y = c1 et + c2 sin(t) + c3 cos(t). Hence y = c1 x + c2 sin(ln(x)) + c3 cos(ln(x)) is the general solution of the given DE.

11.

Exact Equations

The DE p0 (x)y 00 + p1 (x)y 0 + p2 (x)y = q(x) is said to be exact if p0 (x)y 00 + p1 (x)y 0 + p2 (x)y =

d (A(x)y 0 + B(x)y). dx

In this case the given DE is reduced to solving the linear DE Z

A(x)y 0 + B(x)y =

q(x)dx + C

a linear first order DE. The exactness condition can be expressed in operator form as p0 D2 + p1 D + p2 = D(AD + B). d Since dx (A(x)y 0 + B(x)y) = A(x)y 00 + (A0 (x) + B(x))y 0 + B 0 (x)y, the exactness condition holds if and only if A(x), B(x) satisfy

A(x) = p0 (x),

B(x) = p1 (x) − p00 (x),

B 0 (x) = p2 (x).

Since the last condition holds if and only if p01 (x) − p000 (x) = p2 (x), we see that the given DE is exact if and only if p000 − p01 + p2 = 0 in which case p0 (x)y 00 + p1 (x)y 0 + p2 (x)y =

d (p0 (x)y 0 + (p1 (x) − p00 (x))y). dx

Example 3. Solve the DE xy 00 + xy 0 + y = x,

(x > 0).

74

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

This is an exact equation since the given DE can be written d (xy 0 + (x − 1)y) = x. dx Integrating both sides, we get xy 0 + (x − 1)y = x2 /2 + A which is a linear DE. The solution of this DE is left as an exercise.

12.

Reduction of Order

If y1 is a non-zero solution of a homogeneous linear n-th order DE, one can always find a second solution of the form y = C(x)y1 where C 0 (x) satisfies a homogeneous linear DE of order n − 1. Since we can choose C 0 (x) 6= 0 we find in this way a second solution y2 = C(x)y1 which is not a scalar multiple of y1 . In particular for n = 2, we obtain a fundamental set of solutions y1 , y2 . Let us prove this for the second order DE p0 (x)y 00 + p1 (x)y 0 + p2 (x)y = 0. If y1 is a non-zero solution we try for a solution of the form y = C(x)y1 . Substituting y = C(x)y1 in the above we get ³

´

³

´

p0 (x) C 00 (x)y1 +2C 0 (x)y10 +C(x)y100 +p1 (x) C 0 (x)y1 +C(x)y10 +p2 (x)C(x)y1 = 0. Simplifying, we get p0 y1 C 00 (x) + (p0 y10 + p1 y1 )C 0 (x) = 0 since p0 y100 + p1 y10 + p2 y1 = 0. This is a linear first order homogeneous DE for C 0 (x). Note that to solve it we must work on an interval where y1 (x) 6= 0. However, the solution found can always be extended to the places where y1 (x) = 0 in a unique way by the fundamental theorem. The above procedure can also be used to find a particular solution of the non-homogenous DE p0 (x)y 00 + p1 (x)y 0 + p2 (x)y = q(x) from a non-zero solution of p0 (x)y 00 + p1 (x)y 0 + p2 (x)y = 0. Example 4. Solve y 00 + xy 0 − y = 0. Here y = x is a solution so we try for a solution of the form y = C(x)x. Substituting in the given DE, we get C 00 (x)x + 2C 0 (x) + x(C 0 (x)x + C(x)) − C(x)x = 0 which simplifies to xC 00 (x) + (x2 + 2)C 0 (x) = 0.

75

N-TH ORDER DIFFERENTIAL EQUATIONS

Solving this linear DE for C 0 (x), we get C 0 (x) = Ae−x so that

Z

2 /2

/x2

dx

C(x) = A

x2 ex2 /2

+B

Hence the general solution of the given DE is Z

y = A1 x + A2 x

dx x2 ex2 /2

.

Example 5. Solve y 00 + xy 0 − y = x3 ex . By the previous example, the general solution of the associated homogeneous equation is Z

yH = A1 x + A2 x

dx x2 ex2 /2

.

Substituting yp = xC(x) in the given DE we get C 00 (x) + (x + 2/2)C 0 (x) = x2 ex . Solving for C 0 (x) we obtain µ

1 C (x) = 2 x2 /2 A2 + x e 0

where H(x) =

Z

4 x+x2 /2

x e

1 x2 ex2 /2

This gives

Z

C(x) = A1 + A2

¶

dx = A2

Z

x4 ex+x

dx + 2 x ex2 /2

2 /2

1 x2 ex2 /2

+ H(x),

dx.

Z

H(x)dx,

We can therefore take Z

yp = x

H(x)dx,

so that the general solution of the given DE is Z

y = A1 x + A2 x

dx x2 ex2 /2

+ yp (x) = yH (x) + yp (x).

PART (VII): SOME APPLICATIONS OF SECOND ORDER DE’S

13.

(*) Vibration System

We now give an application of the theory of second order DE’s to the description of the motion of a simple mass-spring mechanical system with a damping device. We assume that the damping force is proportional to the velocity of the mass. If there are no external forces we obtain the differential equation d2 x dx + kx = 0 +b 2 dt dt where x = x(t) is the displacement from equilibrium at time t of the mass of m > 0 units, b ≥ 0 is the damping constant and k > 0 is the spring d constant. In operator form with D = dt this DE is, after normalizing, m

µ

¶

b k D + D+ (x) = 0. m m 2

The characteristic polynomial r2 + (b/m)r + k/m has discriminant ∆ = (b2 − 4km)/m2 . If b2 < 4km we have ∆ < 0 and the characteristic polynomial factorizes in the form (r + b/2m)2 + ω 2 with ω=

s

p

4km − b2 /2m =

k − (b/2m)2 . m

In this case the characteristic polynomial has complex roots −b/2m ± iω and the general solution of the DE is y = e−bt/2m (A cos(ωt) + B sin(ωt) = Ce−bt/2m sin(ωt + θ)

77

78

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

√ where C = A2 + B 2 and 0 ≤ θ ≤ 2π defined by cos(θ) = A/C, sin(θ) = B/C. The angle θ is called the phase. In this case we see that the mass oscillates with frequency ω/2π and decreasing amplitude. If b = 0 there is no damping and the mass oscillates with frequency ω/2π and constant amplitude; such motion is called simple harmonic. If b2 ≥ 4km we have ∆ ≥ 0 and so the characteristic polynomial has real roots r1 = −b/2m +

p

b2 − 4km/2m,

r2 = −b/2m −

p

b2 − 4km/2m

which are both negative. If r1 = r2 = r the general solution of our DE is y = Aert + Btert and if r1 6= r2 the general solution is y = Aer1 t + Ber2 t . In both cases y → 0 as t → ∞. In the second case we have what is called over damping and in the first case the over damping is said to be critical. In each the mass returns to its equilibrium position without oscillations. Suppose now that our mass-spring system is subject to an external force so that our DE now becomes m

d2 x dx + kx = F (t). +b 2 dt dt

The function F (t) is called the forcing function and measures the magnitude and direction of the external force. We consider the important special case where the forcing function is harmonic F (f ) = F0 cos(γt),

F0 > 0 a constant.

We also assume that we have under-damping with damping constant b > 0. In this case the DE has a particular solution of the form yp = A1 cos(γt) + A2 sin(γt). Substituting the the DE and simplifying, we get ((k−mγ 2 )A1 +bγA2 ) cos(γt)+(−bγA1 +(k−mγ 2 )A2 ) sin(γt) = F0 cos(γt). Setting the corresponding coefficients on both sides equal, we get (k − mγ 2 )A1 + bγA2 = F0 , −bγA1 + (k − mγ 2 )A2 = 0.

(3.32)

79

N-TH ORDER DIFFERENTIAL EQUATIONS

Solving for A1 , A2 we get A1 =

F0 (k − mγ 2 ) , (k − mγ 2 )2 + b2 γ 2

A2 =

F0 bγ (k − mγ 2 )2 + b2 γ 2

and yp = (k−mγF2 )02 +b2 γ 2 ((k − mγ 2 ) cos(γt) + bγ sin(γt)) F0 sin(γt + φ). =√ 2 2 2 2

(3.33)

(k−mγ ) +b γ

The general solution of our DE is then y = Ce−bt/2m sin(ωt + θ) + p

F0 sin(γt + φ). (k − mγ 2 )2 + b2 γ 2

Because of damping the first term tends to zero and is called the transient part of the solution. The second term, the steady-state part of the solution, is due to the presence of the forcing function F0 cos(γt). It is harmonic with the same frequency γ/2π but is out of phase with it by an angle φ − π/2. The ratio of the magnitudes 1 (k − mγ 2 )2 + b2 γ 2

M (γ) = p

is called the gain factor. The graph of the function M (γ) is called the resonance curve. It has a maximum of 1 q

b q

k m

−

b2 4m2

2

k b when γ = γr = m − 2m 2 . The frequency γr /2π is called the resonance frequency of the system. When γ = γr the system is said to be in resonance with the external force. Note that M (γr ) gets arbitrarily large as b → 0. We thus see that the system is subject to large oscillations if the damping constant is very small and the forcing function has a frequency near the resonance frequency of the system.

The above applies to a simple LRC electrical circuit where the differential equation for the current I is d2 I dI + R + I/C = F (t) 2 dt dt where L is the inductance, R is the resistance and C is the capacitance. The resonance phenomenon is the reason why we can send and receive and amplify radio transmissions sent simultaneously but with different frequencies. L

Chapter 4 SERIES SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS

81

PART (I): SERIES SOLUTIONS NEAR A ORDINARY POINT

A function f (x) of one variable x is said to be analytic at a point x = x0 if it has a convergent power series expansion f (x) =

∞ X

an (x−x0 )n = a0 +a1 (x−x0 )+a2 (x−x0 )2 +· · ·+an (x−x0 )n +· · ·

0

for |x − x0 | < R, R > 0. This point x = x0 is also called ordinary point. Otherwise, f (x) is said to have a singularity at x = x0 . The largest such R (possibly +∞) is called the radius of convergence of the power series. The series converges for every x with |x − x0 | < R and diverges for every x with |x − x0 | > R. There is a formula for R = 1` , where 1 |an+1 | `= , or lim , 1/n n→∞ |an | lim |an | n→∞

if the latter limit exists. The same is true if x, x0 , ai are complex. For example, 1 = 1 − x2 + x4 − x6 + · · · + (−1)n x2n + · · · 1 + x2 for |x| < 1. The radius of convergence of the series is 1. It is also equal to the distance from 0 to the nearest singularity x = i of 1/(x2 + 1) in the complex plane. Power series can be integrated and differentiated within the interval (disk) of convergence. More precisely, for |x − x0 | < R we have f 0 (x) =

∞ X

nan xn−1 =

n=1

∞ X

(n + 1)an+1 xn ,

n=0

83

84

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS Z xX ∞ 0 n=0

an tn =

∞ X

∞ an n+1 X an−1 n x = x n+1 n n=0 n=1

and the resulting power series have R as radius of convergence. If f (x), g(x) are analytic at x = x0 then so is f (x)g(x) and af +bg for any scalars a, b with radii of convergence at least that of the smaller of the radii of convergence the series for f (x), g(x). If f (x) is analytic at x = x0 and f (x0 ) 6= 0 then 1/f (x0 ) is analytic at x = x0 with radius of convergence equal to the distance from x0 to the nearest zero of f (x) in the complex plane. The following theorem shows that linear DE’s with analytic coefficients at x0 have analytic solutions at x0 with radius of convergence as big as the smallest of the radii of convergence of the coefficient functions.

1. 1.1

Series Solutions near a Ordinary Point Theorem

If p1 (x), p2 (x), . . . , pn (x), q(x) are analytic at x = x0 , the solutions of the DE y (n) + p1 (x)y (n−1) + · · · + pn (x)y = q(x) are analytic with radius of convergence ≥ the smallest of the radii of convergence of the coefficient functions p1 (x), p2 (x), . . . , pn (x), q(x). The proof of this result follows from the proof of fundamental existence and uniqueness theorem for linear DE’s using elementary properties of analytic functions and the fact that uniform limits of analytic functions are analytic. Example 1. The coefficients of the DE y 00 + y = 0 are analytic everywhere, in particular at x = 0. Any solution y = y(x) has therefore a series representation y=

∞ X

an xn

n=0

with infinite radius of convergence. We have y0 =

∞ X

(n + 1)an+1 xn ,

n=0

y 00 =

∞ X

(n + 1)(n + 2)an+2 xn .

n=0

Therefore, we have y 00 + y =

∞ X

((n + 1)(n + 2)an+2 + an )xn = 0

n=0

SERIES SOLUTION OF LINEARDIFFERENTIAL EQUATIONS

85

for all x. It follows that (n + 1)(n + 2)an+2 + an = 0 for n ≥ 0. Thus an an+2 = − , for n ≥ 0 (n + 1)(n + 2) from which we obtain a0 a1 a2 a0 a2 = − , a3 = − , a4 = − = , 1·2 2·3 3·4 1·2·3·4 a3 a1 a5 = − = . 4·5 2·3·4·5 By induction one obtains a0 a1 a2n = (−1)n , a2n+1 = (−1)n (2n)! (2n + 1)! and hence that y = a0

∞ X

(−1)n

n=0

∞ X x2n+1 x2n + a1 (−1)n = a0 cos(x) + a1 sin(x). (2n)! (2n + 1)! n=0

Example 2. The simplest non-constant DE is y 00 + xy = 0 which is known as Airy’s equation. Its coefficients are analytic everywhere and so the solutions have a series representation y=

∞ X

an xn

n=0

with infinite radius of convergence. We have y 00 + xy = =

∞ X

(n + 1)(n + 2)an+2 xn +

∞ X

n=0 ∞ X

n=0 ∞ X

n=0

n=1

(n + 1)(n + 2)an+2 xn +

= 2a2 +

∞ X

an xn+1 , an−1 xn ,

(4.1)

((n + 1)(n + 2)an+2 + an−1 )xn = 0

n=1

from which we get a2 = 0, (n + 1)(n + 2)an+2 + an−1 = 0 for n ≥ 1. Since a2 = 0 and an−1 , for n ≥ 1 an+2 = − (n + 1)(n + 2) we have a3 = −

a0 a1 , a4 = − , a5 = 0, 2·3 3·4 a3 a0 a6 = − = , 5·6 2·3·5·6 a4 a1 a7 = − = . 6·7 3·4·6·7

86

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

By induction we get a3n+2 = 0 for n ≥ 0 and a3n = (−1)n a3n+1 = (−1)n

a0 , 2 · 3 · 5 · 6 · · · (3n − 1) · 3n a1 . 3 · 4 · 6 · 7 · · · (3n) · (3n + 1)

Hence y = a0 y1 + a1 y2 with y1 = 1 −

x3 x6 x3n + − · · · + (−1)n +···, 2·3 2·3·5·6 2 · 3 · 5 · 6 · · · (3n − 1) · 3n

y2 = x −

x4 x7 x3n+1 + − · · · (−1)n +···. 3·4 3·4·6·7 3 · 4 · 6 · 7 · · · (3n) · (3n + 1)

For positive x the solutions of the DE y 00 + xy = 0 behave like the solutions to a mass-spring system with variable spring constant. The solutions oscillate for x > 0 with increasing frequency as |x| → ∞. For x < 0 the solutions are monotone. For example, y1 , y2 are increasing functions of x for x ≤ 0.

PART (II): SERIES SOLUTION NEAR A REGULAR SINGULAR POINT

In this lecture we investigate series solutions for the general linear DE a0 (x)y (n) + a1 (x)y (n−1) + · · · + an (x)y = b(x), where the functions a1 , a2 , . . . , an , b are analytic at x = x0 . If a0 (x0 ) 6= 0 the point x = x0 is called an ordinary point of the DE. In this case, the solutions are analytic at x = x0 since the normalized DE y (n) + p1 (x)y (n−1) + · · · + pn (x)y = q(x), where pi (x) = ai (x)/a0 (x), q(x) = b(x)/a0 (x), has coefficient functions which are analytic at x = x0 . If a0 (x0 ) = 0, the point x = x0 is said to be a singular point for the DE. If k is the multiplicity of the zero of a0 (x) at x = x0 and the multiplicities of the other coefficient functions at x = x0 is as big then, on cancelling the common factor (x − x0 )k for x 6= x0 , the DE obtained holds even for x = x0 by continuity, has analytic coefficient functions at x = x0 and x = x0 is an ordinary point. In this case the singularity is said to be removable. For example, the DE xy 00 + sin(x)y 0 + xy = 0 has a removable singularity at x = 0.

2.

Series Solutions near a Regular Singular Point

In general, the solution of a linear DE in a neighborhood of a singularity is extremely difficult. However, there is an important special case where this can be done. For simplicity, we treat the case of the general second order homogeneous DE a0 (x)y 00 + a1 (x)y 0 + a2 (x)y = 0,

(x > x0 ),

with a singular point at x = x0 . Without loss of generality we can, after possibly a change of variable x − x0 = t, assume that x0 = 0. We say

87

88

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

that x = 0 is a regular singular point if the normalized DE y 00 + p(x)y 0 + q(x)y = 0,

(x > 0),

is such that xp(x) and x2 q(x) are analytic at x = 0. A necessary and sufficient condition for this is that lim x2 q(x) = q0

lim xp(x) = p0 ,

x→0

x→0

exist and are finite. In this case xp(x) = p0 + p1 x + · · · + pn xn + · · · ,

x2 q(x) = q0 + q1 x + · · · + qn xn + · · ·

and the given DE has the same solutions as the DE x2 y 00 + x(xp(x))y 0 + x2 q(x)y = 0. This DE is an Euler DE if xp(x) = p0 , x2 q(x) = q0 . This suggests that we should look for solutions of the form r

y=x

Ã∞ X

!

n

an x

=

n=0

∞ X

an xn+r ,

n=0

with a0 6= 0. Substituting this in the DE gives ∞ X

n+r

(n + r)(n + r − 1)an x

n=0

+

Ã∞ X

pn x

n

!Ã ∞ X

n=0

+

Ã∞ X

!

(n + r)an x

n=0

n

qn x

!Ã ∞ X

n=0

n+r

! n+r

an x

=0

n=0

which, on expansion and simplification, becomes a0 F (r)xr +

∞ n X

F (n + r)an + [(n + r − 1)p1 + q1 ]an−1 + · · ·

n=1

o

+(rpn + qn )a0 xn+r = 0, (4.2) where F (r) = r(r − 1) + p0 r + q0 . Equating coefficients to zero, we get r(r − 1) + p0 r + q0 = 0,

(4.3)

the indicial equation, and F (n + r)an = −[(n + r − 1)p1 + q1 ]an−1 − · · · − (rpn + qn )a0 (4.4) for n ≥ 1. The indicial equation (4.3) has two roots: r1 , r2 . Three cases should be discussed separately.

89

SERIES SOLUTION OF LINEARDIFFERENTIAL EQUATIONS

2.1

Case (I): The roots (r1 − r2 6= N )

Two roots do’nt differ by an integer. In this case, the above recursive equation (4.4) determines an uniquely for r = r1 and r = r2 . If an (ri ) is the solution for r = ri and a0 = 1, we obtain the linearly independent solutions y1 = x

r1

Ã∞ X

!

an (r1 )x

n

r2

,

y2 = x

Ã∞ X

n=0

!

n

an (r2 )x

.

n=0

It can be shown that the radius of convergence of the infinite series is the distance to the singularity of the DE nearest to the singularity x = 0. If r1 − r2 = N ≥ 0, the above recursion equations can be solved for r = r1 as above to give a solution y1 = xr1

Ã∞ X

!

an (r1 )xn .

n=0

A second linearly independent solution can then be found by reduction of order. However, the series calculations can be quite involved and a simpler method exists which is based on solving the recursion equation for an (r) as a ratio of polynomials of r. This can always be done since F (n + r) is not the zero polynomial for any n ≥ 0. If an (r) is the solution with a0 (r) = 1 and we let r

y = y(x, r) = x

Ã∞ X

!

n

an (r)x

.

(4.5)

n=0

Thus, we have the following equality with two variables (x, r): x2 y 00 + x2 p(x)y 0 + x2 q(x)y = a0 F (r)xr = (r − r1 )(r − r2 )xr . (4.6)

2.2

Case (II): The roots (r1 = r2 )

In this case, from the equality (4.6) we get x2 y 00 + x2 p(x)y 0 + x2 q(x)y = (r − r1 )2 xr . Differentiating this equation with respect to r, we get µ

x2

∂y ∂r

¶00

µ

+ x2 p(x)

∂y ∂r

¶0

+ x2 q(x)

Setting r = r1 , we find that ∂y y2 = (x, r1 ) = xr1 ∂r

Ã∞ X n=0

∂y = 2(r − r1 ) + (r − r1 )2 xr ln(x). ∂r !

an (r1 )xn ln(x) + xr1

∞ X n=0

a0n (r1 )xn

90

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

= y1 ln(x) + xr1

∞ X

a0n (r1 )xn ,

n=0

a0n (r)

where is the derivative of an (r) with respect to r. This is a second linearly independent solution. Since this solution is unbounded as x → 0, any solution of the given DE which is bounded as x → 0 must be a scalar multiple of y1 .

2.3

Case (III): The roots (r1 − r2 = N > 0)

For this case, we let z(x, r) = (r − r2 )y(x, r). Thus, from the equality (4.6) we get x2 z 00 + x2 p(x)z 0 + x2 q(x)z = (r − r1 )(r − r2 )2 xr . Differentiating this equation with respect to r, we get µ

x

2

∂z ∂r

¶00

µ

∂z + x p(x) ∂r 2

¶0

+ x2 q(x)

∂z = (r − r1 )(r − r2 )2 xr ln(x) ∂r h

i

+(r − r2 ) (r − r2 ) + 2(r − r1 ) xr . Setting r = r2 , we see that y2 = ∂z ∂r (x, r2 ) is a solution of the given DE. Letting bn (r) = (r − r2 )an (r), we have h

i

F (n + r)bn (r) = − (n + r − 1)p1 + q1 bn−1 (r) − · · · −(rpn + qn )b0 (r) and

Ã

y2 = lim

r

r→r2

x ln(x)

∞ X n=0

n

(4.7)

r

bn (r)x + x

∞ X

!

b0n (r)xn

.

(4.8)

n=0

Note that an (r2 ) 6= 0, for n = 1, 2, . . . N − 1. Hence, we have b0 (r2 ) = b1 (r2 ) = b2 (r2 ) = · · · = bN −1 (r2 ) = 0. However, aN (r2 ) = ∞, as F (r2 + N ) = F (r1 ) = 0. Hence, we have bN (r2 ) = lim (r − r2 )an (r) = a < ∞, r→r2

subsequently,

lim xr ln(x)bN (r)xN = axr1 ln(x).

r→r2

Furthermore, F (N + 1 + r2 )bN +1 (r2 ) = F (1 + r1 )bN +1 (r2 ) = −(r1 p1 + q1 )bN (r2 ) − · · · − (r2 pN +1 + qN +1 )b0 (r2 ) = −(r1 p1 + q1 )bN (r2 ) (4.9)

SERIES SOLUTION OF LINEARDIFFERENTIAL EQUATIONS

91

Thus, bN +1 (r2 ) =

(r1 p1 + q1 ) (r1 p1 + q1 ) bN (r2 ) = a = aa1 (r1 ). F (1 + r1 ) F (1 + r1 )

(4.10)

Similarly, we have F (N + 2 + r2 )bN +2 (r2 ) = F (2 + r1 )bN +2 (r2 ) = −[(1 + r1 )p1 + q1 ]bN +1 (r2 ) − (r1 p2 + q2 )bN (r2 ) − · · · − (r2 pN +2 + qN +2 )b0 (r2 ) = −a[(1 + r1 )p1 + q1 ]a1 (r1 ) − a(r1 p2 + q2 ), (4.11) then we obtain bN +2 (r2 ) = −a

[(1 + r1 )p1 + q1 ]a1 (r1 ) + (r1 p2 + q2 ) = aa2 (r1 ).(4.12) F (2 + r1 )

In general, we can write bN +k (r2 ) = aak (r1 ).

(4.13)

Substituting the above results to (4.8), we finally derive r1

y2 = ax

Ã∞ X

! n

an (r1 )x

n=0

= ay1 ln(x) + xr2

Ã∞ X

r2

ln(x) + x

!

Ã∞ X

!

b0n (r2 )xn

n=0

b0n (r2 )xn .

(4.14)

n=0

This gives a second linearly independent solution. The above method is due to Frobenius and is called the Frobenius method. Example 1. The DE 2xy 00 + y 0 + 2xy = 0 has a regular singular point at x = 0 since xp(x) = 1/2 and x2 q(x) = x2 . The indicial equation is 1 1 r(r − 1) + r = r(r − ). 2 2 The roots are r1 = 1/2, r2 = 0 which do not differ by an integer. We have (r + 1)(r + 12 )a1 = 0, (n + r)(n + r − 12 )an = −an−2

for n ≥ 2,

(4.15)

92

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

so that an = −2an−2 /(r + n)(2r + 2n − 1) for n ≥ 2. Hence 0 = a1 = a3 = · · · a2n+1 for n ≥ 0 and 2 a0 , (r + 2)(2r + 3) 2 22 a4 = − a2 = a0 . (r + 4)(2r + 7) (r + 2)(r + 4)(2r + 3)(2r + 7) a2 = −

It follows by induction that 2n (r + 2)(r + 4) · · · (r + 2n) 1 × a0 . (2r + 3)(2r + 4) · · · (2r + 2n − 1)

a2n = (−1)n

(4.16)

Setting, r = 1/2, 0, a0 = 1, we get y1 =

∞ √ X x

x2n , (5 · 9 · · · (4n + 1))n! n=0

y2 =

∞ X

x2n . (3 · 7 · · · · (4n − 1))n! n=0

The infinite series have an infinite radius of convergence since x = 0 is the only singular point of the DE. Example 2. The DE xy 00 + y 0 + y = 0 has a regular singular point at x = 0 with xp(x) = 1, x2 q(x) = x. The indicial equation is r(r − 1) + r = r2 = 0. This equation has only one root x = 0. The recursion equation is (n + r)2 an = −an−1 ,

n ≥ 1.

The solution with a0 = 1 is an (r) = (−1)n

(r +

1)2 (r

1 . + 2)2 · · · (r + n)2

setting r = 0 gives the solution y1 =

∞ X

(−1)n

n=0

xn . (n!)2

Taking the derivative of an (r) with respect to r we get, using a0n (r) = an (r)

d ln [an (r)] dr

SERIES SOLUTION OF LINEARDIFFERENTIAL EQUATIONS

93

(logarithmic differentiation),we get µ

a0n (r) = −

¶

2 2 2 + + ··· + an (r) r+1 r+2 r+n

so that

1

a0n (0) = 2(−1)n 1

+

1 2

+ ··· + (n!)2

1 n

.

Therefore a second linearly independent solution is y2 = y1 ln(x) + 2

∞ X

1

(−1)n 1

n=1

+

1 2

+ ··· + (n!)2

1 n

xn .

The above series converge for all x. Any bounded solution of the given DE must be a scalar multiple of y1 .

PART (III): BESSEL FUNCTIONS

3.

Bessel Equation

In this lecture we study an important class of functions which are defined by the differential equation x2 y 00 + xy 0 + (x2 − ν 2 )y = 0, where ν ≥ 0 is a fixed parameter. This DE is known Bessel’s equation of order ν. This equation has x = 0 as its only singular point. Moreover, this singular point is a regular singular point since xp(x) = 1,

x2 q(x) = x2 − ν 2 .

Bessel’s equation can also be written y 00 +

1 0 ν2 y + (1 − 2 ) = 0 x x

which for x large is approximately the DE y 00 + y = 0 so that we can expect the solutions to oscillate for x large. The indicial equation is r(r − 1) + r − ν 2 = r − ν 2 whose roots are r1 = ν and r2 = −ν. The recursion equations are [(1 + r)2 − ν 2 ]a1 = 0,

[(n + r)2 − ν 2 ]an = −an−2 ,

for n ≥ 2.

The general solution of these equations is a2n+1 = 0 for n ≥ 0 and (−1)n a0 (r + 2 − ν)(r + 4 − ν) · · · (r + 2n − ν) 1 × . (r + 2 + ν)(r + 4 + ν) · · · (r + 2n + ν)

a2n (r) =

95

96

4.

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

The Case of Non-integer ν

If ν is not an integer and ν 6= 1/2, we have the case (I). Two linearly independent solutions of Bessel’s equation Jν (x), J−ν (x) can be obtained by taking r = ±ν, a0 = 1/2ν Γ(ν + 1). Since, in this case, a2n =

(−1)n a0 , 22n n!(r + 1)(r + 2) · · · (r + n)

we have for r = ±ν ∞ X

(−1)n Jr (x) = n!Γ(r + n + 1) n=0

µ ¶2n+r

x 2

.

Recall that the Gamma function Γ(x) is defined for x ≥ −1 by Γ(x + 1) =

Z ∞ 0

e−t tx dt.

For x ≥ 0 we have Γ(x + 1) = xΓ(x), so that Γ(n + 1) = n! for n an integer ≥ 0. We have µ ¶

1 Γ 2

=

Z ∞ 0

−t −1/2

e t

dt = 2

Z ∞ 0

2

e−x dt =

√ π.

The Gamma function can be extended uniquely for all x except for x = 0, −1, −2, . . . , −n, . . . to a function which satisfies the identity Γ(x) = Γ(x)/x. This is true even if x is taken to be complex. The resulting function is analytic except at zero and the negative integers where it has a simple pole. These functions are called Bessel functions of first kind of order ν. As an exercise the reader can show that r

J 1 (x) = 2

5.

2 sin(x), πx

r

J− 1 = 2

2 cos(x). πx

The Case of ν = −m with m an integer ≥ 0

For this case, the first solution Jm (x) can be obtained as in the last section. As examples, we give some such solutions as follows: The Case of m = 0: J0 (x) =

∞ X (−1)n

22n (n!)2 n=0

x2n

SERIES SOLUTION OF LINEARDIFFERENTIAL EQUATIONS

97

The case m = 1: ∞ 1 xX (−1)n J1 (x) = y1 (x) = x2n . 2 2 n=0 22n n!(n + 1)!

To derive the second solution, one has to proceed differently. For ν = 0 the indicial equation has a repeated root, we have the case (II). One has a second solution of the form y2 = J0 (x) ln(x) +

∞ X

a02n (0)x2n

n=0

where a2n (r) =

(−1)n . (r + 2)2 (r + 4)2 · · · (r + 2n)2

It follows that

µ

a02n (r) 1 1 1 = −2 + + ··· a2n r+2 r+4 r + 2n so that

¶

µ

a02n (0) = − 1 + where we have defined

¶

1 1 + ··· + a2n (0) = −hn a2n (0), 2 n µ

¶

1 1 hn = 1 + + · · · + . 2 n Hence y2 = J0 (x) ln(x) +

∞ X (−1)n+1 hn 2n x . 2n 2

n=0

2 (n!)

Instead of y2 , the second solution is usually taken to be a certain linear combination of y2 and J0 . For example, the function Y0 (x) =

i 2h y2 (x) + (γ − ln 2)J0 (x) , π

where γ = lim (hn − ln n) ≈ 0.5772, is known as the Weber function n→∞ of order 0. The constant γ is known as Euler’s constant; it is not known whether γ is rational or not. If ν = −m, with m > 0, the the roots of the indicial equation differ by an integer, we have the case (III). Then one has a solution of the form y2 = aJm (x) ln(x) +

∞ X n=0

b02n (−m)x2n+m

98

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

where b2n (r) = (r + m)a2n (r) and a = b2m (−m). In the case m = 1 we have a0 = 1, a0 a = b2 (−1) = − , 2 b0 (r) = (r − r2 )a0 and for n ≥ 1, b2n (r) =

(−1)n a0 . (r + 3)(r + 5) · · · (r + 2n − 1)(r + 3)(r + 5) · · · (r + 2n + 1)

Subsequently, we have

b00 (r) = a0

and for n ≥ 1, µ

b02n (r)

=−

1 1 1 1 + + ··· + + r+3 r+5 r + 2n − 1 r + 3 ¶ 1 1 + ··· + b2n (r). + r+5 r + 2n + 1

From here, we obtain b00 (−1) = a0 b02n (−1) = −1 2 (hn + hn−1 )b2n (−1)

(4.17)

(n ≥ 1),

where b2n (−1) =

(−1)n a0 . 22n (n − 1)!n!

So that "

∞ X −1 1 (−1)n+1 (hn + hn−1 ) 2n y2 = y1 (x) ln(x) + 1+ x 2 x 22n+1 (n − 1)!n! n=1

"

∞ X (−1)n+1 (hn + hn−1 ) 2n 1 1+ x = −J1 (x) ln(x) + x 22n+1 (n − 1)!n! n=1

#

#

where, by convention, h0 = 0, (−1)! = 1. The Weber function of order 1 is defined to be Y1 (x) =

i 4h − y2 (x) + (γ − ln 2)J1 (x) . π

The case m > 1 is slightly more complicated and will not be treated here.

SERIES SOLUTION OF LINEARDIFFERENTIAL EQUATIONS

99

The second solutions y2 (x) of Bessel’s equation of order m ≥ 0 are unbounded as x → 0. It follows that any solution of Bessel’s equation of order m ≥ 0 which is bounded as x → 0 is a scalar multiple of Jm . The solutions which are unbounded as x → 0 are called Bessel functions of the second kind. The Weber functions are Bessel functions of the second kind.

Chapter 5 LAPLACE TRANSFORMS

101

PART (I): LAPLACE TRANSFORM AND ITS INVERSE

1.

Introduction

We begin our study of the Laplace Transform with a motivating example. This example illustrates the type of problem that the Laplace transform was designed to solve. A mass-spring system consisting of a single steel ball is suspended from the ceiling by a spring. For simplicity, we assume that the mass and spring constant are 1. Below the ball we introduce an electromagnet controlled by a switch. Assume that, we on, the electromagnet exerts a unit force on the ball. After the ball is in equilibrium for 10 seconds the electromagnet is turned on for 2π seconds and then turned off. Let y = y(t) be the downward displacement of the ball from the equilibrium position at time t. To describe the motion of the ball using techniques previously developed we have to divide the problem into three parts: (I) 0 ≤ t < 10; (II) 10 ≤ t < 10 + 2π; (III) 10 + 2π ≤ t. The initial value problem determining the motion in part I is y 00 + y = 0,

y(0) = y 0 (0) = 0.

The solution is y(t) = 0, 0 ≤ t < 10. Taking limits as t → 10 from the left, we find y(10) = y 0 (10) = 0. The initial value problem determining the motion in part II is y 00 + y = 1,

y(10) = y 0 (10) = 0.

The solution is y(t) = 1 − cos(t − 10), 10 ≤ t < 2π + 10. Taking limits as t → 10 + 2π from the left, we get y(10 + 2π) = y 0 (10 + 2π) = 0. The initial value problem for the last part is y 00 + y = 0,

y(10 + 2π) = y 0 (10 + 2π) = 0

103

104

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

which has the solution y(t) = 0, 10 + 2π ≤ t. Putting all this together, we have 0, 0 ≤ t < 10, 1 − cos(t − 10), 10 ≤ t < 10 + 2π, y(t) = 0, 10 + 2π ≤ t. The function y(t) is continuous with continuous derivative y 0 (t) =

0,

0 ≤ t < 10, sin(t − 10), 10 ≤ t < 10 + 2π, 0, 10 + 2π ≤ t.

However the function y 0 (t) is not differentiable at t = 10 and t = 10+2π. In fact 0, 0 ≤ t < 10, 00 cos(t − 10), 10 < t < 10 + 2π, y (t) = 0, 10 + 2π < t. The left hand and right hand limits of f 00 (t) at t = 10 are 0 and 1 respectively. At t = 10+2π they are 1 and 0. If we extend y 00 (t) by using the left-hand or righthand limits at 10 and 10+2π the resulting function is not continuous. Such a function with only jump discontinuities is said to be piecewise continuous. If we try to write the differential equation of the system we have y 00 + y = f (t) =

0,

1, 0,

0 ≤ t < 10, 10 ≤ t < 10 + 2π, 10 + 2π ≤ t.

Here f (t) is piecewise continuous and any solution would also have y 00 piecewise continuous. By a solution we mean any function y = y(t) satisfying the DE for those t not equal to the points of discontinuity of f (t). In this case we have shown that a solution exists with y(t), y 0 (t) continuous. In the same way, one can show that in general such solutions exist using the fundamental theorem. What we want to describe now is a mechanism for doing such problems without having to divide the problem into parts. This mechanism is the Laplace transform.

2. 2.1

Laplace Transform Definition

Let f (t) be a function defined for t ≥ 0. The function f (t) is said to be piecewise continuous if (1) f (t) converges to a finite limit f (0+ ) as t → 0+

105

LAPLACE TRANSFORMS

(2) for any c > 0, the left and right hand limits f (c− ), f (c+ ) of f (t) at c exist and are finite. (3) f (c− ) = f (c+ ) = f (c) for every c > 0 except possibly a finite set of points or an infinite sequence of points converging to +∞. Thus the only points of discontinuity of f (t) are jump discontinuities. The function is said to be normalized if f (c) = f (c+ ) for every c ≥ 0. The Laplace transform F (s) = L{f (t)} is the function of a new variable s defined by F (s) =

Z ∞ 0

e

−st

f (t)dt =

lim

Z N

N →+∞ 0

e−st f (t)dt.

An important class of functions for which the integral converges are the functions of exponential order. The function f (t) is said to be of exponential order if there are constants a, M such that |f (t)| ≤ M eat for all t. the solutions of constant coefficient homogeneous DE’s are all of exponential order. The convergence of the improper integral follows from Z N Z N 1 e−(s−a) −st |e f (t)|dt ≤ M e−(s−a)t dt = − s−a s−a 0 0 which shows that the improper integral converges absolutely when s > a. It shows that F (s) → 0 as s → ∞. The calculation also shows that 1 L{eat } = s−a for s > a. Setting a = 0, we get L{1} =

2.2

1 s

for s > 0.

Basic Properties and Formulas

The above holds when f (t) is complex valued and s = σ + iτ is complex. The integral exists in this case for σ > a. For example, this yields 1 1 , L{e−it } = . L{eit } = s−i s+i

2.2.1

Linearity of the transform

L{af (t) + bf (t)} = aL{f (t)} + bL{g(t)}. Using this linearity property of the Laplace transform and using sin(t) = it (e − e−it )/2i, cos(t) = (eit + e−it )/2, we find 1 L{sin(bt)} = 2i

µ

1 1 − s − bi s + bi

¶

=

s2

b , + b2

106

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

1 L{cos(bt)} = 2

µ

1 1 + s − bi s + bi

¶

=

s2

s , + b2

for s > 0.

2.2.2 Formula (I) The following two identities follow from the definition of the Laplace transform after a change of variable. L{eat f (t)}(s) = L{f (t)}(s − a), 1 L{f (bt)}(s) = L{f (t)}(s/b). b Using the first of these formulas, we get L{eat sin(bt)} =

b , (s − a)2 + b2

L{eat cos(t)} =

s−a . (s − a)2 + b2

2.2.3 Formula (II) The next formula will allow us to find the Laplace transform for all the functions that are annihilated by a constant coefficient differential operator. dn L{tn f (t)}(s) = (−1)n n L{f (t)}(s). ds For n = 1 this follows from the definition of the Laplace transform on differentiating with respect s and taking the derivative inside the integral. The general case follows by induction. For example, using this formula, we obtain using f (t) = 1 L{tn }(s) = −

dn 1 n! = n+1 . n ds s s

With f (t) = sin(t) and f (t) = cos(t) we get L{t sin(bt)}(s) = − L{t cos(bt)}(s) = −

d b 2bs = 2 , 2 2 ds s + b (s + b2 )2

s s2 − b2 1 2b2 d = = − . ds s2 + b2 (s2 + b2 )2 s2 + b2 (s2 + b2 )2

2.2.4 Formula (III) The next formula shows how to compute the Laplace transform of f 0 (t) in terms of the Laplace transform of f (t). L{f 0 (t)}(s) = sL{f (t)}(s) − f (0).

107

LAPLACE TRANSFORMS

This follows from L{f 0 (t)}(s) =

Z ∞ 0

=s

¯∞ ¯

e−st f 0 (t)dt = e−st f (t)¯

Z ∞ 0

0

+s

Z ∞ 0

e−st f (t)dt

e−st f (t)dt − f (0)

(5.1)

since e−st f (t) converges to 0 as t → +∞ in the domain of definition of the Laplace transform of f (t). To ensure that the first integral is defined, we have to assume f 0 (t) is piecewise continuous. Repeated applications of this formula give L{f (n) (t)}(s) = sn L{f (t)}(s) − sn−1 f (0) − sn−2 f 0 (0) − · · · − f n−1 (0). The following theorem is important for the application of the Laplace transform to differential equations.

3. 3.1

Inverse Laplace Transform Theorem:

If f (t), g(t) are normalized piecewise continuous functions of exponential order then L{f (t)} = L{g(t)} implies f = g.

3.2

Definition

If F (s) is the Laplace of the normalized piecewise continuous function f (t) of exponential order then f (t) is called the inverse Laplace transform of F (s). This is denoted by F (s) = L−1 {f (t)}. Note that the inverse Laplace transform is also linear. Using the Laplace transforms we found for t sin(bt), t cos(bt) we find ½

L−1 and

½

L−1

s 2 (s + b2 )2

1 (s2 + b2 )2

¾

=

¾

=

1 t sin(bt), 2b

1 1 sin(bt) − 2 t cos(bt). 2b3 2b

PART (II): SOLVE DE’S WITH LAPLACE TRANSFORMS

4.

Solve IVP of DE’s with Laplace Transform Method

In this lecture we will, by using examples, show how to use Laplace transforms in solving differential equations with constant coefficients.

4.1

Example 1

Consider the initial value problem y 00 + y 0 + y = sin(t),

y(0) = 1, y 0 (0) = −1.

Step 1 Let Y (s) = L{y(t)}, we have L{y 0 (t)} = sY (s) − y(0) = sY (s) − 1, L{y 00 (t)} = s2 Y (s) − sy(0) − y 0 (0) = s2 Y (s) − s + 1. Taking Laplace transforms of the DE, we get (s2 + s + 1)Y (s) − s =

s2

1 . +1

Step 2 Solving for Y (s), we get Y (s) =

s 1 + . s2 + s + 1 (s2 + s + 1)(s2 + 1)

109

110

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

Step 3 Finding the inverse laplace transform. ½

y(t) = L−1 {Y (s)} = L−1

s s2 + s + 1

¾

½

¾

1 . (s2 + s + 1)(s2 + 1)

+ L−1

Since s2

s s s + 1/2 √ = = 2 +s+1 (s + 1/2) + 3/4 (s + 1/2)2 + ( 3/2)2 √ 1 3/2 √ −√ 2 3 (s + 1/2) + ( 3/2)2

we have ½

L

−1

s 2 s +s+1

¾

√ √ 1 = e−t/2 cos( 3 t/2) − √ e−t/2 sin( 3 t/2). 3

Using partial fractions we have 1 As + B Cs + D = 2 + 2 . (s2 + s + 1)(s2 + 1) s +s+1 s +1 Multiplying both sides by (s2 + 1)(s2 + s + 1) and collecting terms, we find 1 = (A + C)s3 + (B + C + D)s2 + (A + C + D)s + B + D. Equating coefficients, we get A + C = 0, B + C + D = 0, A + C + D = 0, B + D = 1, from which we get A = B = 1, C = −1, D = 0, so that ½

L

−1

1 2 (s + s + 1)(s2 + 1)

¾

½

¾

s =L + L−1 2 s +s+1 ½ ¾ s −1 −L . s2 + 1 −1

½

1 2 s +s+1

¾

Since ½

L

−1

1 2 s +s+1

we obtain

¾

√ 2 = √ e−t/2 sin( 3 t/2), 3

½ −1

L

s 2 s +1

√ y(t) = 2e−t/2 cos( 3 t/2) − cos(t).

¾

= cos(t)

111

LAPLACE TRANSFORMS

4.2

Example 2

As we have known, a higher order DE can be reduced to a system of DE’s. Let us consider the system dx = −2x + y, dt dy = x − 2y dt

(5.2)

with the initial conditions x(0) = 1, y(0) = 2.

Step 1 Taking Laplace transforms the system becomes sX(s) − 1 = −2X(s) + Y (s), sY (s) − 2 = X(s) − 2Y (s),

(5.3)

where X(s) = L{x(t)}, Y (s) = L{y(t)}.

Step 2 Solving for X(s), Y (s). The above linear system of equations can be written in the form: (s + 2)X(s) − Y (s) = 1, −X(s) + (s + 2)Y (s) = 2.

(5.4)

The determinant of the coefficient matrix is s2 + 4s + 3 = (s + 1)(s + 3). Using Cramer’s rule we get X(s) =

s2

s+4 , + 4s + 3

Y (s) =

s2

2s + 5 . + 4s + 3

Step 3 Finding the inverse Laplace transform. Since s+4 3/2 1/2 = − , (s + 1)(s + 3) s+1 s+3

2s + 5 3/2 1/2 = + , (s + 1)(s + 3) s+1 s+3

we obtain 3 1 x(t) = e−t − e−3t , 2 2

3 1 y(t) = e−t + e−3t . 2 2

The Laplace transform reduces the solution of differential equations to a partial fractions calculation. If F (s) = P (s)/Q(s) is a ratio of

112

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

polynomials with the degree of P (s) less than the degree of Q(s) then F (s) can be written as a sum of terms each of which corresponds to an irreducible factor of Q(s). Each factor Q(s) of the form s − a contributes the terms Ar A1 A1 + ··· + + 2 s − a (s − a) (s − a)r where r is the multiplicity of the factor s − a. Each irreducible quadratic factor s2 + as + b contributes the terms A1 s + B1 A2 s + B2 Ar s + Br + 2 + ··· + 2 2 2 s + as + b (s + as + b) (s + as + b)r where r is the degree of multiplicity of the factor s2 + as + b.

PART (III): FURTHER STUDIES OF LAPLACE TRANSFORM

5. 5.1

Step Function Definition ½

uc (t) =

5.2

0 1

t < c, t ≥ c.

Laplace transform of unit step function L{uc (t)} =

e−cs . s

One can derive L{uc (t)f (t − c)} = e−cs F (s).

6. 6.1

Impulse Function Definition

Let dτ (t) = It follows that I(τ ) =

½ 1

2τ

0

Z ∞ −∞

|t| < τ, |t| ≥ τ.

dτ (t)dt = 1.

Now, consider the limit, ½

δ(t) = lim dτ (t) = τ →0

113

0 ∞

t 6= 0, t = 0,

114

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

which is called the Dirac δ-function. Evidently, the Dirac δ-function has the following properties: 1

Z ∞ −∞

2

Z B A

3

Z B A

6.2

δ(t)dt = 1. ½

δ(t − c)dt =

0 1

½

δ(t − c)f (t)dt =

¯ (A, B), c∈ c ∈ (A, B).

0 f (c)

¯ (A, B), c∈ c ∈ (A, B).

Laplace transform of unit step function L{δ(t − c)} =

½ −cs e

0

c > 0, c < 0.

One can derive L{δ(t − c)f (t)} = e−cs f (c),

7. 7.1

(c > 0).

Convolution Integral Theorem

Given L{f (t)} = F (s), one can derive

L{g(t)} = G(s),

L−1 {F (s) · G(s)} = f (t) ∗ g(t),

where f (t) ∗ g(t) =

Z τ

is called the convolution integral.

0

f (t − τ )g(τ )dτ

Chapter 6 (*) SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS

115

(*) PART (I): INTRODUCTION OF SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS

In this and the following lecture we will give an introduction to systems of differential equations. For simplicity, we will limit ourselves to systems of two equations with two unknowns. The techniques introduced can be used to solve systems with more equations and unknowns. As a motivational example, consider the the following problem.

1.

Mathematical Formulation of a Practical Problem

Two large tanks, each holding 24 liters of brine, are interconnected by two pipes. Fresh water flows into tank A a the rate of 6 L/min, and fluid is drained out tank B at the same rate. Also, 8 L/min of fluid are pumped from tank A to tank B and 2 L/min from tank B to tank A. The solutions in each tank are well stirred sot that they are homogeneous. If, initially, tank A contains 5 in solution and Tank B contains 2 kg, find the mass of salt in the tanks at any time t. To solve this problem, let x(t) and y(t) be the mass of salt in tanks A and B respectively. The variables x, y satisfy the system dx 1 = −1 3 x + 12 y, dt (6.1) dy = 13 x − 13 y. dt The first equation gives y = 12 dx dt + 4x. Substituting this in the second equation and simplifying, we get 1 d2 x 2 dx + + x = 0. 2 dt 3 dt 12

117

118

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

The general solution of this DE is x = c1 e−t/2 + c2 e−t/6 . −t/2 + 2c e−t/6 . Thus the general This gives y = 12 dx 2 dt + 4x = −2c1 e solution of the system is

x = c1 e−t/2 + c2 e−t/6 , y = −2c1 e−t/2 + 2c2 e−t/6 .

(6.2)

These equations can be written in matrix form as µ

X=

x y

¶

µ

= c1 e

−t/2

1 −2

µ ¶

¶

+ c2 e

−t/6

1 2

.

Using the initial condition x(0) = 5, y(0) = 2, we find c1 = 2, c2 = 3. Geometrically, these equations are the parametric equations of a curve (trajectory of the DE) in the xy-plane (phase plane of the DE). As t → ∞ we have (x(t), y(t)) → (0, 0). The constant solution x(t) = y(t) = 0 is called an equilibrium solution of our system. This solution is said to be asymptotically stable if the general solution converges to it as t → ∞. A system is called stable if the trajectories are all bounded as t → ∞. Our system can be written in matrix form as dX dt = AX where µ

A=

−1/3 1/12 1/3 −1/3

¶

X.

The 2 × 2 matrix A is called the matrix of the system. The polynomial 2 1 r2 − tr(A)r + det(A) = r2 + r + 3 12 where tr(A) is the trace of A (sum of diagonal entries) and det(A) is the determinant of A is called the characteristic polynomial of A. Notice that this polynomial is the characteristic polynomial of the differential equation for x. The equations µ

A µ

¶

1 −2

¶

−1 = 2

µ ¶

µ

1 −2

¶

µ ¶

,

A

1 2

−1 = 6

µ ¶

1 2

1 1 identify and as eigenvectors of A with eigenvalues −1/2 −2 2 and −1/6 respectively. More generally, a non-zero column vector X is an eigenvector of a square matrix A with eigenvalue r if AX = rX or , equivalently, (rI − A)X = 0. The latter is a homogeneous system

(*) SYSTEMS OF LINEAR DIFFERENTIALEQUATIONS

119

of linear equations with coefficient matrix rI − A. Such a system has a non-zero solution if and only if det(rI − A) = 0. Notice that det(rI − A) = r2 − (a + d)r + ad − bc is the characteristic polynomial of A. If, in the above mixing problem, brine at a concentration of 1/2 kg/L was pumped into tank A instead of pure water the system would be dx 1 = −1 3 x + 12 y + 3, dt dy = 13 x − 13 y, dt

(6.3)

a non-homogeneous system. Here an equilibrium solution would be x(t) = a, y(t) = b where (a, b) was a solution of −1 1 3 x + 12 y 1 1 3x − 3y

= −3, = 0.

(6.4)

In this case a = b = 12. The variables x∗ = x − 12, y ∗ = y − 12 then satisfy the homogeneous system dx∗ 1 ∗ ∗ = −1 3 x + 12 y , dt∗ dy = 13 x∗ − 31 y ∗ . dt

(6.5)

Solving this system as above for x∗ , y ∗ we get x = x∗ + 12, y = y ∗ + 12 as the general solution for x, y.

2.

(2 × 2) System of Linear Equations

We now describe the solution of the system dX dt = AX for an arbitrary 2 × 2 matrix A. In practice, one can use the elimination method or the eigenvector method but we shall use the eigenvector method as it gives an explicit description of the solution. There are three main cases depending on whether the discriminant ∆ = tr(A)2 − 4 det(A) of the characteristic polynomial of A is > 0, < 0, = 0.

2.1

Case 1: ∆ > 0

In this case the roots r1 , r2 of the characteristic polynomial are real and unequal, say r1 < r2 . Let Pi be an eigenvector with eigenvalue ri .

120

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

Then P1 is not a scalar multiple of P2 and so the matrix P with columns P1 , P2 is invertible. After possibly replacing P2 by −P2 , we can assume that det(P ) > 0. The equation µ

AP = P shows that

r1 0 0 r2

µ

P

−1

AP =

r1 0 0 r2

¶

¶

. µ

If we make the change of variable X = P U with U =

¶

u , our system v

becomes

dU dU = AP U or = P −1 AP U. dt dt Hence, our system reduces to the uncoupled system P

du = r1 u, dt

dv = r2 v dt

which has the general solution u = c1 er1 t , v = c2 er2 t . Thus the general solution of the given system is X = P U = uP1 + vP2 = c1 er1 t P1 + c2 er2 t P2 . Since tr(A) = r1 + r2 , det(A) = r1 r2 , we see that x(t), y(t) = (0, 0) is an asymptotically stable equilibrium solution if and only if tr(A) < 0 and det(A) > 0. The system is unstable if det(A) < 0 or det(A) ≥ 0 and tr(A) ≥ 0.

2.2

Case 2: ∆ < 0

In this case the roots of the characteristic polynomial are complex numbers q r = α ± iω = tr(A)/2 ± i ∆/4. The corresponding eigenvectors of A are (complex) scalar multiples of µ

1 σ ± iτ

¶

where σ = (α − a)/b, τ = ω/b. If X is a real solution we must have X = V + V with 1 V = (c1 + ic2 )eαt (cos(ωt) + i sin(ωt)) 2

µ

1 σ + iτ

¶

.

121

(*) SYSTEMS OF LINEAR DIFFERENTIALEQUATIONS

It follows that µ αt

X = e (c1 cos(ωt) − c2 sin(ωt))

1 σ

¶

µ αt

+ e (c1 sin(ωt) + c2 cos(ωt))

0 τ

¶

.

The trajectories are spirals if tr(A) 6= 0 and ellipses if tr(A) = 0. The system is asymptotically stable if tr(A) < 0 and unstable if tr(A) > 0.

2.3

Case 3: ∆ = 0

Here the characteristic polynomial has only one root r. If A = rI the system is dx dy = rx, = ry. dt dt which has the general solution x = c1 ert , y = c2 ert . Thus the system is asymptotically stable if tr(A) < 0, stable if tr(A) = 0 and unstable if tr(A) > 0. Now suppose A 6= rI. If P1 is an eigenvector with eigenvalue r and P2 is chosen with (A − rI)P1 6= 0, the matrix P with columns P1 , P2 is invertible and µ ¶ r 1 −1 P AP = . 0r Setting as before X = P U we get the system du = ru + v, dt

dv = rv dt

which has the general solution u = c1 ert + c2 tert , v = c2 ert . Hence the given system has the general solution X = uP1 + vP2 = (c1 ert + c2 tert )P1 + c2 ert P2 . The trajectories are asymptotically stable if tr(A) < 0 and unstable if tr(A) ≥ 0. A non-homogeneous system dX dt = AX + B having an equilibrium solution x(t) = x1 , y(t) = y1 can be solved by introducing new variables x∗ = x − x1 , y ∗ = y − y1 . Since AX ∗ + B = 0 we have dX ∗ = AX ∗ , dt a homogeneous system which can be solved as above.

(*) PART (II): EIGENVECTOR METHOD

In this lecture we will apply the eigenvector method to the solution of a second order system of the type arising in the solution of a mass-spring system with two masses. The system we will consider consists of two masses with mass m1 , m2 connected by a spring with spring constant k2 . The first mass is attached to the ceiling of a room by a spring with spring constant k1 and the second mass is attached to the floor by a spring with spring constant k3 at a point immediately below the point of attachment to the ceiling. Assume that the system is under tension and in equilibrium. If x1 (t), x1 (t) are the displacements of the two masses from their equilibrium position at time t, the positive direction being upward, then the motion of the system is determined by the system d2 x1 = −k1 x1 − k2 (x1 − x2 ) = −(k1 + k2 )x1 + k2 x2 , 2 dt 2 d x2 m2 2 = k2 (x1 − x2 ) − k3 x2 = k2 x1 − (k2 + k3 )x2 . dt

m1

The system can be written in matrix form µ

X=

x1 x2

¶

µ

,

A=

d2 X dt2

(6.6)

= AX where

−(k1 + k2 )/m1 k2 /m2 k2 /m1 −(k2 + k3 )/m2

¶

.

The characteristic polynomial of A is "

#

m2 (k1 + k2 ) + m1 (k2 + k3 ) (k1 + k2 )(k2 + k3 ) k22 r + r+ − . m1 m2 m1 m2 m1 m2 2

123

124

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

The discriminant of this polynomial is ∆=

½h i2 1 m (k + k ) + m (k + k ) 2 1 2 1 2 3 m21 m22 ¾

−4(k1 + k2 )(k2 + k3 )m1 m2 + 4k22 m1 m2 =

(m2 (k1 + k2 ) − m1 (k2 + k3 m21 m22

))2

(6.7)

+ 4m1 m2 k2

2

> 0.

Hence the eigenvalues of A are real, distinct and negative since the trace of A is negative while the determinant is positive. Let r1 > r2 be the eigenvalues of A and let µ

P1 =

1 s1

¶

µ

,

P2 =

1 s2

¶

be (normalized) eigenvectors with eigenvalues r1 , r2 respectively. We have m1 r1 + k1 + k2 m1 r2 + k1 + k2 s1 = , s2 = k2 k2 and, if P is the matrix with columns P1 , P2 , we have µ

P −1 AP =

r1 0 0 r2

¶

. µ

If we make a change of variables X = P Y with Y = d2 Y = dt2

µ

r1 0 0 r2

¶

y1 , we have y2

¶

so that our system in the new variables y1 , y2 is d2 y1 dt2 d2 y2 dT 2

= r1 y1 = r2 y2 .

(6.8)

Setting ri = −ωi2 with ωi > 0, this uncoupled system has the general solution y1 = A1 sin(ω1 t) + B1 cos(ω1 t),

y2 = A2 sin(ω2 t) + B2 cos(ω2 t).

Since X = P Y = y1 P1 + y2 P2 , we obtain the general solution X = (A1 sin(ω1 t) + B1 cos(ω1 t))P1 + (A2 sin(ω2 t) + B2 cos(ω2 t))P2 . The two solutions with Y (0) = Pi are of the form X = (A sin(ωi t) + B cos(ωi t))Pi =

p

A2 + B 2 sin(ωi t + θi )Pi .

125

(*) SYSTEMS OF LINEAR DIFFERENTIALEQUATIONS

These motions are simple harmonic with frequencies ωi /2π and are called the fundamental motions of the system. Since any motion of the system is the sum (superposition) of two such motions any periodic motion of the system must have a period which is an integer multiple of both the fundamental periods 2π/ω1 , 2π/ω2 . This happens if and only if ω1 /ω2 is a rational number. If X 0 (0) = 0, the fundamental motions are of the form X = Bi cos(ωi t)Pi and if X(0) = 0 they are of the form X = Ai sin(ωi t)Pi . These four motions are a basis for the solution space of the given system. The motion is completely determined once X(0) and X 0 (0) are known since µ

X(0) = P Y (0) = P

B1 B2

¶

µ

,

X 0 (0) = P Y 0 (0) = P

ω1 A1 ω2 A2

¶

.

As a particular example, consider the case where m1 = m2 = m and k1 = k2 = k3 = k. The system is symmetric and A=

k m

µ

−2 1 1 −2

¶

,

a symmetric matrix. The characteristic polynomial is r2 + 4

k2 k k k r + 3 2 = (r + )(r + 3 ). m m m m

The eigenvalues are pr1 = −k/m,pr2 = −3k/m. The fundamental frequencies are ω1 = k/m, ω2 = 3k/m. The normalized eigen-vectors are µ ¶ µ ¶ 1 1 P1 = , P2 = . 1 −1 The fundamental motions with X 0 (0) = 0 are q

X = A cos( k/m t)

µ ¶

1 1

q

,

X = A cos( 3k/m t)

µ

1 −1

¶

.

√ Since the ratio of the fundamental frequencies is 3, an irrational number, theses are the only two periodic motions of the mass-spring system where the masses are displaced and then let go. Odds and Ends

126

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

If y = f (x) is a solution of the autonomous DE y n = f (y, y 0 , . . . , y n−1 ) then so is y = f (x + a) for any real number a. If the DE is linear and homogeneous with fundamental set y1 , y2 , . . . , yn then we must have identities of the form y1 (x + a) = c2 y2 + c3 y3 + · · · + cn yn . For example, consider the DE y 00 + y = 0. Here sin(x), cos(x) is a fundamental set so we must have an identity of the form sin(x + a) = A sin(x) + B cos(x). Differentiating, we get cos(x + a) = A cos(x) − B sin(x). Setting x = 0 in these two equations we find A = cos(a), B = sin(a). We obtain in this way the addition formulas for the sine and cosine functions: sin(x + a) = sin(x) cos(a) + sin(a) cos(x), cos(x + a) = cos(x) cos(a) − sin(x) sin(a). The numerical methods for solving DE’s can be extended to systems virtually without change. In this way we can get approximate solutions for higher order DE’s. For more details consult the text (Chapter 5).

Appendix A ASSIGNMENTS AND SOLUTIONS

127

128

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

Assignment 2B: due Thursday, September 21, 2000

1 Find the solution of the initial value problem yy 0 = x(y 2 − 1)4/3 ,

y(0) = b > 0.

What is its interval of definition? (Your answer will depend on the value of b.) Sketch the graph of the solution when b = 1/2 and when b = 2 2 Find the general solution of the differential equation dy = y + e2x y 3 . dx

3 Solve the initial value problem dy x y = + , dx y x

y(1) = −4.

4 Solve the initial value problem (ex − 1)

dy + yex + 1 = 0, dx

y(1) = 1.

Solutions for Assignment 2B

1 Separating variables and integrating we get

Z

yy 0 dx x2 = + C1 4/3 2 − 1)

(y 2

from which, on making the change of variables u = y 2 , we get 1 2

Z (u − 1)−4/3 du =

x2 + C1 . 2

Integrating and simplifying, we get (u − 1)−1/3 = C − x2 /3

with C = −2C1 /3.

Hence (y 2 − 1)−1/3 = C − x2 /3. Then y(0) = b gives C = (b2 − 1)−1/3 . Since b > 0 we must have r 1 . y = 1+ (C − x2 )3

APPENDIX A: ASSIGNMENTS AND SOLUTIONS

129

If b < 1 then √ y is defined for all x while, if b > 1, the solution y is defined only for |x| < 3C. 2 This is a Bernoulli equation. To solve it, divide both sides by y 3 and make the change of variables u = 1/y 2 . This gives u0 = −2u − 2e2x after multiplication by −2. We now have a linear equation whose general solution is u = −e2x /2 + Ce−2x . This gives a 1-paramenter family of solutions ±1

y= p

Ce−2x − e2x /2

= p

±ex C − e4x /2

of the original DE. Given (x0 , y0 ) with y0 6= 0 there is a unique value of C such that the solution satisfies y(x0 ) = y0 . It is not the general solution as it omits the solution y = 0. Thus the general solution is comprised of the functions y = 0,

±ex

y= p

C − e4x /2

.

3 This is a homogeneous equation. Setting u = y/x, we get xu0 + u = 1/u + u. This gives xu0 = 1/u, a separable equation from which we get uu0 = 1/x. Integrating, we get u2 /2 = ln |x| + C1 and hence y 2 = x2 ln(x2 ) + Cx2 with C = 2C1 . For y(1) = −4 we must have C = 16 and p y = −x ln(x2 ) + 16, x > 0. 4 This is a linear equation which is also exact. The general solution is F (x, y) = C where ∂F ∂F = yex − 1, = ey − 1. ∂x ∂y Integrating the first equation partially with respect to x we get F (x, y) = yex + x + φ(y) from which and hence

∂F ∂y

= ex + φ0 (y) = ey − 1 which gives φ(y) = −y (up to a constant) F (x, y) = yex + x − y = C.

For y(1) = 1 we must have C = e and so the solution is y=

e−x , ex − 1

(x > 0).

130

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

Assignment 3B: due Thursday, September 28, 2000

1 One morning it began to snow very hard and continued to snow steadily through the day. A snowplow set out at 8:00 A.M. to clear a road, clearing 2 miles by 11:00 A.M. and an additional mile by 1:00 P.M. At what time did it start snowing. (You may assume that it was snowing at a constant rate and that the rate at which the snowplow could clear the road was inversely proportional to the depth of the snow.) 2 Find, in implicit form, the general solution of the differential equation y 3 + 4yex + (2ex + 3y 2 )y 0 = 0. Given x0 , y0 , is it always possible to find a solution such that y(x0 ) = y0 ? If so, is this solution unique? Justify your answers.

APPENDIX A: ASSIGNMENTS AND SOLUTIONS

131

Solution for Assignment 3B

1 Let x be the distance travelled by the snow plow in t hours with t = 0 at 8 AM. Then if it started snowing at t = −b we have a dx = . dt t+b The solution of this DE is x = a ln(t + b) + c. Since x(0) = 0, x(3) = 2, x(5) = 3, we have a ln b + c = 0, a ln(3 + b) = 2, a ln(5 + b) + c = 3 from which a ln

3+b = 2, b

a ln

5+b = 1. 3+b

2 2 Hence (3 + b)/b = (5 + b)2 /(3 + b) √ from which b − 2b − 27 = 0. The positive root of this equation is b = 1 + 2 7 ≈ 6.29 hours. Hence it started snowing at 1 : 42 : 36 AM.

2 The DE y 3 + 4yex + (2ex + 3y 2 )y 0 = 0 has an integrating factor µ = ex . The solution in implicit form is 2e2x y + y 3 ex = C. There is a unique solution with y(x0 ) = y0 for any x0 , y0 by the fundamental existence and uniqueness theorem since the coefficient of y 0 in the DE is never zero and hence f (x, y) =

−y 3 − 4yex 2ex + 3y 2

and its partial derivative fy are continuously differentiable on R2 . Alternately, since the partial derivative of y 3 ex + 2ye2x with respect to y is never zero, the implicit function theorem guarantees the existence of a unique function y = y(x), with y(x0 ) = y0 and defined in some neighborhood of x0 , which satisfies the given DE.

132

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

Assignment 4B: due Tuesday, October 24, 2000

1 (a) Show that the differential equation M + N y 0 = 0 has an integrating factor which is a function of z = x + y only if and only if ∂M ∂y

−

∂N ∂x

M −N is a function of z only. (b) Use this to solve the differential equation x2 + 2xy − y 2 + (y 2 + 2xy − x2 )y 0 = 0. 2 Solve the differential equations (a) xy 00 = y 0 + x, 00

(b) y(y − 1)y + y

(x > 0); 02

= 0.

3 Solve the differential equations (a) y 000 − 3y 0 + 2y = ex ; (b) y (iv) − 2y 000 + 5y 00 − 8y 0 + 4y = sin(x). 4 Show that the functions sin(x), sin(2x), sin(3x) are linearly independent. Find a homogeneous linear ODE having these functions as part of a basis for its solution space. Show that it is not possible to find such an ODE with these functions as a basis for its solution space.

APPENDIX A: ASSIGNMENTS AND SOLUTIONS

133

Solutions to Assignment 4B 1 (a) Suppose that M + N y 0 = 0 has an integrating factor u which is a function of ) ) z = x + y. Then ∂(uM = ∂(uN gives ∂y ∂x u(

∂N ∂u ∂u ∂M − )=N −N . ∂y ∂x ∂x ∂y

By the chain rule we have ∂u du ∂z du = = , ∂x dz ∂x dz so that

∂M ∂y

−

∂u du ∂z du = = , ∂y dz ∂y dz

∂N ∂x

−1 du = , M −N u dz which is a function of z. Conversely, suppose that ∂M ∂y

−

∂N ∂x

M −N

= f (z),

with z = x + y. Now define u = u(z) to be a solution of the linear DE du = −f (z)u. Then dz ∂M − ∂N −1 du ∂y ∂x = , M −N u dz ) ) = ∂(uN , i.e., that u is an integrating factor of which is equivalent to ∂(uM ∂y ∂x 0 M + N y which is a function of z = x + y only.

(b) For the DE x2 + 2xy − y 2 + (y 2 + 2xy − x2 )y 0 = 0 we have ∂M ∂y

−

∂N ∂x

M −N

=

2 2 = . x+y z

If we define

R u=e

−2dz/z

= e−2 ln z = 1/z 2 = 1/(x + y)2

then u is an integrating factor so that there is a function F (x, y) with ∂F x2 + 2xy − y 2 = uM = , ∂x (x + y)2

∂F y 2 + 2xy − x2 = uN = . ∂y (x + y)2

Integrating the first DE partially with respect to x, we get

Z F (x, y) =

(1 −

2y 2 2y 2 dx = x + + φ(y). 2 (x + y) x+y

Differentiating this with respect to y and using the second DE, we get y 2 + 2xy − x2 ∂F 2y 2 + 4xy = = + φ0 (y) (x + y)2 ∂y (x + y)2

134

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS so that φ0 (y) = −1 and hence φ(y) = −y (up to a constant). Thus F (x, y) = x +

2y 2 x2 + y 2 −y = . x+y x+y

Thus the general solution of the DE is F (x, y) = C or x + y = 0 which is the only solution that was missed by the integrating factor method. The first solution is the family of circles x2 + y 2 − Cx − Cy = 0 passing through the origin and center on the line y = x. Through any point 6= (0, 0) there passes a unique solution. 2 (a) The dependent variable y is missing from the DE xy 00 = y 0 + x. Set w = y 0 so that w0 = y 00 . The DE becomes xw0 = w + x which is a linear DE with general solution w = x ln(x) + C1 x. Thus y 0 = x ln(x) + C1 which gives y=

x2 x2 x2 x2 x2 ln(x) − + C1 + C2 = ln(x) + A +B 2 4 2 2 2

with A, B arbitrary constants. 2

(b) The independent variable x is missing DE y(y −1)+y 0 = 0. Note that y = C is a solution. We assume that y 6= C. Let w = y 0 . Then y 00 =

dw dw dy dw = =w dx dy dx dy

so that the given DE becomes y(y − 1) dw = −w after dividing by w which is dy not zero. Separating variables and integrating, we get

Z

dw =− w

Z

dy y(y − 1)

which gives ln |w| = ln |y| − ln |y − 1| + C1 . Taking exponentials, we get w=

Ay . y−1

Since w = y 0 we have a separable equation for y. Separating variables and integrating, we get y − ln |y| = Ax + B1 . Taking exponentials, we get ey /y = BeAx with A arbitrary and B 6= 0 as an implicit definition of the non-constant solutions. 3 (a) The associated homogeneous DE is (D3 − 3Dy + 2)(y) = 0. Since D3 − 3D + 2 = (D − 1)2 (D − 2) this DE has the general solution yh = (A + Bx)ex + Ce2x . Since the RHS of the original DE is killed by D − 1, a particular solution yp of it satisfies the DE (D − 1)3 (D − 2) = 0 and so must be of the form (A + Bx + Ex2 )ex + Ce2x . Since we can delete the terms which are solutions of the homogeneous DE, we can take yp = Ex2 ex . Substituting this in the original DE, we find E = 1/6 so that the general solution is y = yh + yp = (A + Bx)E x + Ce2x + x2 ex /6.

135

APPENDIX A: ASSIGNMENTS AND SOLUTIONS

(b) The associated homogeneous DE is (D4 − 2D3 + 5D2 − 8D + 4)(y) = 0. Since D4 − 2D3 + 5D2 − 8D + 4 = (D − 1)2 (D + 4) this DE has general solution yh = (A + Bx)ex + E sin(2x) + F cos(2x). A particular solution yp is a solution of the DE (D2 + 1)(D − 1)2 (D2 + 4)(y) = 0 so that there is a particular solution of the form C1 cos(x) + C2 sin(x). Substituting in the original equation, we find C1 = 1/6, C2 = 0. Hence y = yh + yp = (A + Bx)ex + E sin(2x) + F cos(2x) +

1 cos(x) 6

is the general solution. 4 (a)

Ã W (sin(x), sin(2x), sin(3x)) =

sin(x) sin(2x) sin(3x) cos(x) 2 cos(2x) 3 cos(3x) − sin(x) −4 sin(2x) −9 sin(3x)

!

so that W (π/2) = −16 6= 0. Hence sin(x), sin(2x), sin(3x) are linearly independent. (b) The DE (D2 + 1)(D2 + 4)(D2 + 9)(y) = 0 has basis sin(x), sin(2x) sin(3x) cos(x), cos(2x) cos(3x) and the given functions are part of it. (c) Since the Wronskian of the given functions is zero at x = 0 it cannot be a fundamental set for a necessarily third order linear DE.

136

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

Assignment 5B: due Thursday, Oct. 24, 2002

1 Find the general solution of the differential equation y 00 + 4y 0 + 4y = e−2x ln(x),

(x > 0).

√ √ 2 Given that y1 = cos(x)/ x, y2 = sin(x)/ x are linearly independent solutions of the differential equation x2 y 00 + xy 0 + (x2 − 1/4)y = 0,

(x > 0),

find the general solution of the equation x2 y 00 + xy 0 + (x2 − 1/4)y = x5/2 ,

(x > 0).

3 Find the general solution of the equation x2 y 00 + 3xy 0 + y = 1/x ln(x),

(x > 0).

4 Find the general solution of the equation (1 − x2 )y 00 − 2xy 0 + 2y = 0,

(−1 < x < 1)

given that y = x is a solution. 5 Find the general solution of the equation xy 00 + xy 0 + y = x,

(x > 0).

APPENDIX A: ASSIGNMENTS AND SOLUTIONS

137

Assignment 5 Solutions 1 The differential equation in operator form is (D+2)2 (y) = e−2x ln(x). Multiplying both sides by e2x and using the fact that D + 2 = e−2x De2x , we get D2 (e2x y) = ln(x). Hence 3 1 e2x y = x2 ln x − x2 + Ax + B 2 4 from which we get y = Axe2x + Be2x + 12 x2 e2x ln x − 34 x2 e−2x . Variation of parameters could also have been used but the solution would have been longer. √ √ 2 The given functions y1 = cos x/ x, y2 = sin x/ x are linearly independent and hence a fundamental set of solutions for the DE y 00 +

1 0 1 y + (1 − 2 )y = 0. x 4x

We only have to find a particular y solution of the normalized DE y 00 +

√ 1 0 1 y + (1 − 2 )y = x. x 4x

Using variation of parameters there is a solution of the form y = uy1 + vy2 with u0 y1 + v 0 y2 = 0, √ u0 y10 + v 0 y20 = x.

(A.1)

By Crammer’s Rule we have

u0 = ¯

¯ ¯ ¯

v0 = ¯

¯ ¯ ¯

¯ ¯0 ¯√ ¯ x

¯ ¯ ¯ ¯

sin √x x − sin x+2x √ sin x 2x x cos x sin √ √x x x − cos x−2x √ sin x − sin x+2x √ sin x 2x x 2x x

¯ ¯ ¯ ¯

¯ ¯ ¯ ¯

cos √ x 0 x √ − cos x−2x √ sin x x 2x x cos x sin x √ √ x x − cos x−2x √ sin x − sin x+2x √ sin x 2x x 2x x

¯ = −x sin x ¯ ¯ ¯

¯ = x cos x ¯ ¯ ¯

so that u = x cos x − sin x, v = x sin x + cos x and √ sin x cos x y = (x cos x − sin x) √ + (x sin x + cos x) √ = x. x x Hence the general solution of the DE x2 y 00 + xy 0 + (x2 − 1/4)y = x5/2 is sin x √ cos x y = A √ + B √ + x. x x 3 This is an Euler equation. So we make the change of variable x = et . The given DE becomes (D(D − 1) + 3D + 1)(y) = e−t /t

138

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

where D = DE is

d . dt

Since D(D − 1) + 3D + 1 = D2 + 2D + 1 = (D + 1)2 , the given (D + 1)2 (y) = e−t /t.

Multiplying both sides by et and using et (D + 1) = Det , we get D2 (et y) = 1/t from which et y = t ln t + At + B and y = Ate−t + Be−t + te−t ln t = A

B ln x ln x + + ln(ln(x)), x x x

the general solution of the given DE. 4 Using reduction of order, we look for a solution of the form y = xv. Then y 0 = xv 0 + v, y 00 = xv 00 + 2v 0 and (1 − x2 )(xv 00 + 2v 0 ) − 2x(xv 0 + v) + 2xv = 0 which simplifies to v 00 +

2 − 4x2 0 v =0 x − x3

which is a linear DE for v 0 . Hence

R

v0 = e Since

2−4x2 x3 −x

dx

.

2 − 4x2 −2 −1 1 = + + , x3 − x x x+1 1−x

we have

1 1 1 1 1 + − . x 2x−1 2x+1 and the general solution of the given DE is

v 0 = 1/x2 (1 − x)(x + 1) = − Hence v = − x1 +

1 2

ln

1+x 1−x

y = Ax + B(−1 +

x 1+x ln . 2 1−x

5 This DE is exact and can be written in the form d (xy 0 + (x − 1)y) = x dx so that xy 0 + (x − 1)y = x2 /2 + C. This is a linear DE. Normalizing, we get y 0 + (1 − 1/x)y = x/2 + C/x. An integrating factor for this equation is ex /x. d ex ( y) = ex /2 + Cex /x2 , dx x ex y = ex /2 + C x

Z

y = x/2 + Cxe

−x

Z

ex dx + D, x2

ex dx + Dxe−x . x2

APPENDIX A: ASSIGNMENTS AND SOLUTIONS

139

Assignment 7B: due Thursday, November 21, 2000

For each of the following differential equations show that x = 0 is a regular singular point. Also, find the indicial equation and the general solution using the Frobenius method. 1 9x2 y 00 + 9xy 0 + (9x − 1)y = 0. 2 xy 00 + (1 − x)y 0 + y = 0. 3 x(x + 1)y 00 + (x + 5)y 0 − 4y = 0.

140

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

Solutions for Assignment 7(b) 1 The differential equation in normal form is y 00 + p(x)y 0 + q(x)y = y 00 +

1 0 y + x

³

1 1 − 2 x 9x

´ y=0

so that x = 0 is a singular point. This point is a regular singular point since x2 q(x) = −

xp(x) = 1,

1 +x 9

are analytic at x = 0. The indicial equation is r(r − 1) + r − 1/9 = 0 so that r2 −1/9 = 0, i.e., r = ±1/3. Using the method of Frobenius, we look for a solution of the form y=

∞ X

an xn+r .

n=0

Substituting this into the differential equation x2 y 00 + x2 p(x)y 0 + x2 q(x)y = 0, we get (r2 − 1/9)a0 xr +

∞ X

(((n + r)2 − 1/9)an + an−1 )xn+r = 0.

n=1

In addition to r = ±1/3, we get the recursion equation an = −

an−1 9an−1 =− (n + r)2 − 1/9 (3n + 3r − 1)(3n + 3r + 1)

for n ≥ 1. If r = 1/3, we have an = −3an−1 /n(3n + 2) and an =

(−1)n 3n a0 . n!5 · 8 · · · (3n + 2)

Taking a0 = 1, we get the solution y1 = x1/3

∞ X n=0

(−1)n 3n a0 xn . n!5 · 8 · · · (3n + 2)

Similarly for r = −1/3, we get the solution y2 = x−1/3

∞ X n=0

(−1)n 3n a0 xn . n!1 · 4 · · · (3n − 2)

The general solution is y = Ay1 + By2 . 2 The differential equation in normal form is y 00 + p(x)y 0 + q(x)y = y 00 + (

1 1 − 1)y 0 + = 0 x x

so that x = 0 is a singular point. This point is a regular singular point since xp(x) = 1 − x,

x2 q(x) = x

APPENDIX A: ASSIGNMENTS AND SOLUTIONS

141

are analytic at x = 0. The indicial equation is r(r − 1) + r = 0 so that r2 = 0, i.e., r = 0. Using the method of Frobenius, we look for a solution of the form y=

∞ X

an xn+r .

n=0

Substituting this into the differential equation x2 y 00 + x2 p(x)y 0 + x2 q(x)y = 0, we get r2 a0 xr +

∞ X

((n + r)2 an − (n + r − 2)an−1 )xn+r = 0.

n=0

This yields the recursion equation an = Hence an (r) =

n+r−2 an−1 , (n + r)2

(n ≥ 1).

(r − 1)r(r + 1) · · · (r + n − 2) a0 . (r + 1)2 (r + 2)2 · · · (r + n)2

Taking r = 0, a0 = 1, we get the solution y1 = 1 − x. To get a second solution we compute a0n (0). Using logarithmic differentiation, we get a0n (r) = an (r)(

1 1 1 2 2 2 + + ··· + − − − ··· − ). r−1 r n+r−2 r+1 r+2 r+n

Hence a01 (0) = 3a0 and a0n (r) = an (r)/r + an (r)bn (r) for n ≥ 2. Setting r = 0, we get for n ≥ 2 (−1) · 1 · 2 · · · · (n − 2) a0n (0) = a0 (n!)2 from which an = −(n − 2)!a0 /(n!)2 for n ≥ 2. Taking a0 = 1, we get as second solution y2 = y1 ln(x) + 3x −

∞ X (n − 2)! n=2

(n!)2

xn = y1 ln(x) + 4x − 1 + y1 .

The general solution is then y = Ay1 + B(y1 ln(x) + 4x − 1).

142

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

Assignment 8B: due Thursday, November 23, 2000

1 (a) Compute the Laplace transforms of the functions t2 sin(t),

t2 cos(t).

(b) Find the inverse Laplace transforms of the functions s , (s2 + 1)3

1 . (s2 + 1)3

2 Using Laplace transforms, solve the initial value problem y iv − y = sin(t),

y(0) = y 0 (0) = 1, y 00 (0) = y 000 (0) = −1.

3 Using Laplace transforms, solve the system dx dt dy dt

= −2x + 3y, =x−y

(A.2)

with the initial conditions x(0) = 1, y(0) = −1. 4 Using Laplace transforms, solve the initial value problem y 00 + 3y 0 + 2y = f (t),

y(0) = y 0 (0) = 0,

where f (t) = { 1 ,

0 ≤ t < 1, − 1,

1 ≤ t < π, sin(t),

π ≤ t.

143

APPENDIX A: ASSIGNMENTS AND SOLUTIONS

Solutions for Assignment 8(b) 1 (a) d −2s 1 = 2 , ds s2 + 1 (s + 1)2 2 s −1 1 2 d s = 2 = 2 − 2 L{t cos(t)} = − ds s2 + 1 (s + 1)2 s +1 (s + 1)2 d −2s 6 8 L{t2 sin(t)} = − = 2 − 2 , ds (s2 + 1)2 (s + 1)2 (s + 1)3 8s 2s L{t2 cos(t)} = − 2 + 2 . (s + 1)3 (s + 1)2 L{t sin(t)} = −

(b)

s } = −t2 cos(t)/8 + t sin(t)/8, (s2 + 1)3 1 L−1 { 2 } = 3 sin(t)/8 − 3t cos(t)/8 − t2 sin(t)/8. (s + 1)3 L−1 {

2 If Y (s) = L{y(t)}, we have (s4 − 1)Y (s) − s3 − s2 + s + 1 =

1 . s2 +1

Hence

((s + 1)(s2 − 1) 1 + (s4 − 1)(s2 + 1) s4 − 1 ((s + 1) 1 = 4 + 2 (s − 1)(s2 + 1) s +1 1 1 1 1 3 1 1 1 s = − + − + 2 . 8s−1 8s+1 4 s2 + 1 2 (s2 + 1)2 s +1

Y (s) =

y(t) = et /8 − e−t /8 + sin(t)/2 + cos(t) + t cos(t)/4. 3 If X(s) = L{x(t)}, Y (s) = L{y(t)} we have sX(s) − 1 = −2X(s) + 3Y (s), Hence we have

sY (s) + 1 = X(s) − Y (s).

(s − 2) , (s2 + 3s − 1) (−1 − s) Y (s) = 2 (s + 3s − 1) X(s) =

and

³ X(s) =

√ 7 13 26

Y (s) = −

³ x(t) =

³√

³√

³ 1 √ s+(3+ 13)/2

+

´

13 26

√ 7 13 26

y(t) = −

´ + 12 + 12

´

1 √ s+(3+ 13)/2 √ −(3+ 13)t/2

+ 12 e

13 26

´

+

√ −(3+ 13)t/2

+ 12 e

³√

³ + +

´

√ −7 13 26 13 26

+

13 26

√ 1 , s−( 13−3)/2

´

√ −7 13 26

³√

1 2

+ 1 2

+

´ 1 2

e

´

+

1 2

√ 1 , 13−3)/2 √ ( 13−3)t/2

s−(

e

,

√ ( 13−3)t/2

4 We have y 00 + 3y 0 + 2y = 1 − 2u1 (t) + (sin(t) + 1)uπ (t). If Y (s) = L{y(t)} (s2 + 3s + 2)Y (s) =

e−s 1 −2 + e−πs 2 s

³

´

−1 1 + , +1 s

s2

144

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

since sin(t + π) = − sin(t). Hence Y (s) =

Since

1 −2 + e−s s(s + 1)(s + 2) s(s + 1)(s + 2) µ ¶ −1 1 −πs +e + . (s2 + 1)(s + 1)(s + 2) s(s + 1)(s + 2)

1 1 1 1 = − + , s(s + 1)(s + 2) 2s s+1 2(s + 2) 1 1 1 − 3s 1 = − + , (s2 + 1)(s + 1)(s + 2) 2(s + 1) 5(s + 2) 10(s2 + 1)

we have

³

´

1 1 −1 2 1 1 − + + + − e−s 2s s + 1 2(s + 2) s s + 1 s + 2 ¶ µ 1 3 7 1 3s + − + − − e−πs . 2s 2(s + 1) 10(s + 2) 10(s2 + 1) 10(s2 + 1) Y (s) =

and

³

´

1 e−2t − e−t + + − 1 + 2e1−t − e2−2t u1 (t) 2 µ 2 π−t ¶ sin(t) 3 cos(t) 1 3e 7e2π−2t + − + + − uπ (t). 2 2 10 10 10 y(t) =

Hence

y(t) =

1 1 − e−t + e−2t , 2 2 − 1 + (2e − 1)e−t + ( 1 − e2 )e−2t , 2

2

1 7 2π −2t 3 e )e (2e − 1 − eπ )e−t + ( − e2 + 2 2 10 1 3 +

10

sin(t) −

10

cos(t),

0 ≤ t < 1, 1 ≤ t < π,

π ≤ t.

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

JIAN-JUN XU AND JOHN LABUTE Department of Mathematics and Statistics, McGill University

Kluwer Academic Publishers Boston/Dordrecht/London

Contents

1. INTRODUCTION 1 Definitions and Basic Concepts 1.1 Ordinary Differential Equation (ODE) 1.2 Solution 1.3 Order n of the DE 1.4 Linear Equation: 1.5 Homogeneous Linear Equation: 1.6 Partial Differential Equation (PDE) 1.7 General Solution of a Linear Differential Equation 1.8 A System of ODE’s 2 The Approaches of Finding Solutions of ODE 2.1 Analytical Approaches 2.2 Numerical Approaches 2. FIRST ORDER DIFFERENTIAL EQUATIONS 1 Linear Equation 1.1 Linear homogeneous equation 1.2 Linear inhomogeneous equation 2 Separable Equations. 3 Logistic Equation 4 Fundamental Existence and Uniqueness Theorem 5 Bernoulli Equation: 6 Homogeneous Equation: 7 Exact Equations. 8 Theorem. 9 Integrating Factors. v

1 1 1 1 1 2 2 2 3 3 4 4 4 5 7 7 8 11 13 14 15 16 19 20 21

vi

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

10

11 12 13 14 15 16

17 18 19 20 21

Change of Variables. 10.1 y 0 = f (ax + by), b 6= 0 dy a1 x + b1 y + c1 10.2 = dx a2 x + b2 y + c2 10.3 Riccatti equation: y 0 = p(x)y + q(x)y 2 + r(x) Orthogonal Trajectories. Falling Bodies with Air Resistance Mixing Problems Heating and Cooling Problems Radioactive Decay Definitions and Basic Concepts 16.1 Directional Field 16.2 Integral Curves 16.3 Autonomous Systems 16.4 Equilibrium Points Phase Line Analysis Bifurcation Diagram Euler’s Method Improved Euler’s Method Higher Order Methods

3. N-TH ORDER DIFFERENTIAL EQUATIONS 1 Theorem of Existence and Uniqueness (I) 1.1 Lemma 2 Theorem of Existence and Uniqueness (II) 3 Theorem of Existence and Uniqueness (III) 3.1 Case (I) 3.2 Case (II) 4 Linear Equations 4.1 Basic Concepts and General Properties 5 Basic Theory of Linear Differential Equations 5.1 Basics of Linear Vector Space 5.1.1 Isomorphic Linear Transformation 5.1.2 Dimension and Basis of Vector Space 5.1.3 (*) Span and Subspace 5.1.4 Linear Independency 5.2 Wronskian of n-functions

23 23 23 24 25 27 27 28 29 31 31 31 31 31 32 32 37 38 38 43 46 46 47 47 49 50 50 50 51 51 51 52 52 52 53

vii

Contents

6

7

8 9 10 11 12 13

5.2.1 Definition 5.2.2 Theorem 1 5.2.3 Theorem 2 The Method with Undetermined Parameters 6.1 Basic Equalities (I) 6.2 Cases (I) ( r1 > r2 ) 6.3 Cases (II) ( r1 = r2 ) 6.4 Cases (III) ( r1,2 = λ ± iµ) The Method with Differential Operator 7.1 Basic Equalities (II). 7.2 Cases (I) ( b2 − 4ac > 0) 7.3 Cases (II) ( b2 − 4ac = 0) 7.4 Cases (III) ( b2 − 4ac < 0) 7.5 Theorems The Differential Operator for Equations with Constant Coefficients The Method of Variation of Parameters Euler Equations Exact Equations Reduction of Order (*) Vibration System

4. SERIES SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS 1 Series Solutions near a Ordinary Point 1.1 Theorem 2 Series Solutions near a Regular Singular Point 2.1 Case (I): The roots (r1 − r2 6= N ) 2.2 Case (II): The roots (r1 = r2 ) 2.3 Case (III): The roots (r1 − r2 = N > 0) 3 Bessel Equation 4 The Case of Non-integer ν 5 The Case of ν = −m with m an integer ≥ 0 5. LAPLACE TRANSFORMS 1 Introduction 2 Laplace Transform 2.1 Definition

53 54 54 57 57 58 59 60 61 61 62 62 63 64 67 68 71 73 74 77 81 84 84 87 89 89 90 95 96 96 101 103 104 104

viii

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

3

4

5

6

7

2.2 Basic Properties and Formulas 2.2.1 Linearity of the transform 2.2.2 Formula (I) 2.2.3 Formula (II) 2.2.4 Formula (III) Inverse Laplace Transform 3.1 Theorem: 3.2 Definition Solve IVP of DE’s with Laplace Transform Method 4.1 Example 1 4.2 Example 2 Step Function 5.1 Definition 5.2 Laplace transform of unit step function Impulse Function 6.1 Definition 6.2 Laplace transform of unit step function Convolution Integral 7.1 Theorem

105 105 106 106 106 107 107 107 109 109 111 113 113 113 113 113 114 114 114

6. (*) SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS 1 Mathematical Formulation of a Practical Problem 2 (2 × 2) System of Linear Equations 2.1 Case 1: ∆ > 0 2.2 Case 2: ∆ < 0 2.3 Case 3: ∆ = 0

115 117 119 119 120 121

Appendices ASSIGNMENTS AND SOLUTIONS

127 127

Chapter 1 INTRODUCTION

1. 1.1

Definitions and Basic Concepts Ordinary Differential Equation (ODE)

An equation involving the derivatives of an unknown function y of a single variable x over an interval x ∈ (I).

1.2

Solution

Any function y = f (x) which satisfies this equation over the interval (I) is called a solution of the ODE. For example, y = e2x is a solution of the ODE y 0 = 2y and y = sin(x2 ) is a solution of the ODE xy 00 − y 0 + 4x3 y = 0.

1.3

Order n of the DE

An ODE is said to be order n, if y (n) is the highest order derivative occurring in the equation. The simplest first order ODE is y 0 = g(x). The most general form of an n-th order ODE is F (x, y, y 0 , . . . , y (n) ) = 0 with F a function of n + 2 variables x, u0 , u1 , . . . , un . The equations xy 00 + y = x3 ,

y 0 + y 2 = 0,

1

y 000 + 2y 0 + y = 0

2

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

are examples of ODE’s of second order, first order and third order respectively with respectively F (x, u0 , u1 , u2 ) = xu2 + u0 − x3 , F (x, u0 , u1 ) = u1 + u20 , F (x, u0 , u1 , u2 , u3 ) = u3 + 2u1 + u0 .

1.4

Linear Equation:

If the function F is linear in the variables u0 , u1 , . . . , un the ODE is said to be linear. If, in addition, F is homogeneous then the ODE is said to be homogeneous. The first of the above examples above is linear are linear, the second is non-linear and the third is linear and homogeneous. The general n-th order linear ODE can be written an (x)

1.5

dn y dn−1 y dy + a0 (x)y = b(x). + a (x) + · · · + a1 (x) n−1 n n−1 dx dx dx

Homogeneous Linear Equation:

The linear DE is homogeneous, if and only if b(x) ≡ 0. Linear homogeneous equations have the important property that linear combinations of solutions are also solutions. In other words, if y1 , y2 , . . . , ym are solutions and c1 , c2 , . . . , cm are constants then c1 y1 + c2 y2 + · · · + cm ym is also a solution.

1.6

Partial Differential Equation (PDE)

An equation involving the partial derivatives of a function of more than one variable is called PED. The concepts of linearity and homogeneity can be extended to PDE’s. The general second order linear PDE in two variables x, y is a(x, y)

∂2u ∂2u ∂2u ∂u + c(x, y) + b(x, y) + d(x, y) 2 2 ∂x ∂x∂y ∂y ∂x ∂u +e(x, y) + f (x, y)u = g(x, y). ∂y

Laplace’s equation

∂2u ∂2u + 2 =0 ∂x2 ∂y

is a linear, homogeneous PDE of order 2. The functions u = log(x2 +y 2 ), u = xy, u = x2 − y 2 are examples of solutions of Laplace’s equation. We will not study PDE’s systematically in this course.

3

INTRODUCTION

1.7

General Solution of a Linear Differential Equation

It represents the set of all solutions, i.e., the set of all functions which satisfy the equation in the interval (I). For example, the general solution of the differential equation y 0 = 3x2 is y = x3 + C where C is an arbitrary constant. The constant C is the value of y at x = 0. This initial condition completely determines the solution. More generally, one easily shows that given a, b there is a unique solution y of the differential equation with y(a) = b. Geometrically, this means that the one-parameter family of curves y = x2 + C do not intersect one another and they fill up the plane R2 .

1.8

A System of ODE’s y10 = G1 (x, y1 , y2 , . . . , yn ) y20 = G2 (x, y1 , y2 , . . . , yn ) .. . yn0 = Gn (x, y1 , y2 , . . . , yn )

An n-th order ODE of the form y (n) = G(x, y, y 0 , . . . , y n−1 ) can be transformed in the form of the system of first order DE’s. If we introduce dependant variables y1 = y, y2 = y 0 , . . . , yn = y n−1 we obtain the equivalent system of first order equations y10 = y2 , y20 = y3 , .. .

(1.1)

yn0 = G(x, y1 , y2 , . . . , yn ). For example, the ODE y 00 = y is equivalent to the system y10 = y2 , y20 = y1 .

(1.2)

In this way the study of n-th order equations can be reduced to the study of systems of first order equations. Some times, one called the latter as the normal form of the n-th order ODE. Systems of equations arise in the study of the motion of particles. For example, if P (x, y) is the position of a particle of mass m at time t, moving in a plane under the action of the force field (f (x, y), g(x, y), we have 2

m ddt2x = f (x, y), 2 m ddt2y = g(x, y).

(1.3)

4

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

This is a second order system. The general first order ODE in normal form is y 0 = F (x, y). If F and ∂F ∂y are continuous one can show that, given a, b, there is a unique solution with y(a) = b. Describing this solution is not an easy task and there are a variety of ways to do this. The dependence of the solution on initial conditions is also an important question as the initial values may be only known approximately. The non-linear ODE yy 0 = 4x is not in normal form but can be brought to normal form 4x y0 = . y by dividing both sides by y.

2. 2.1

The Approaches of Finding Solutions of ODE Analytical Approaches

Analytical solution methods: finding the exact form of solutions; Geometrical methods: finding the qualitative behavior of solutions; Asymptotic methods: finding the asymptotic form of the solution, which gives good approximation of the exact solution.

2.2

Numerical Approaches

Numerical algorithms — numerical methods; Symbolic manipulators — Maple, MATHEMATICA, MacSyma. This course mainly discuss the analytical approaches and mainly on analytical solution methods.

Chapter 2 FIRST ORDER DIFFERENTIAL EQUATIONS

5

PART (I): LINEAR EQUATIONS

In this lecture we will treat linear and separable first order ODE’s.

1.

Linear Equation

The general first order ODE has the form F (x, y, y 0 ) = 0 where y = y(x). If it is linear it can be written in the form a0 (x)y 0 + a1 (x)y = b(x) where a0 (x), a( x), b(x) are continuous functions of x on some interval (I). To bring it to normal form y 0 = f (x, y) we have to divide both sides of the equation by a0 (x). This is possible only for those x where a0 (x) 6= 0. After possibly shrinking I we assume that a0 (x) 6= 0 on (I). So our equation has the form (standard form) y 0 + p(x)y = q(x) with p(x) = a1 (x)/a0 (x) and q(x) = b(x)/a0 (x), both continuous on (I). Solving for y 0 we get the normal form for a linear first order ODE, namely y 0 = q(x) − p(x)y.

1.1

Linear homogeneous equation

Let us first consider the simple case: q(x) = 0, namely, dy + p(x)y = 0. dx With the chain law of derivative, one may write d y 0 (x) = ln [y(x)] = −p(x), y dx

7

8

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

integrating both sides, we derive Z

ln y(x) = − or

p(x)dx + C,

y = C1 e−

where C, as well as C1 =

1.2

eC ,

R

p(x)dx

,

is arbitrary constant.

Linear inhomogeneous equation

We now consider the general case: dy + p(x)y = q(x). dx We multiply the both sides of our differential equation with a factor µ(x) 6= 0. Then our equation is equivalent (has the same solutions) to the equation µ(x)y 0 (x) + µ(x)p(x)y(x) = µ(x)q(x). We wish that with a properly chosen function µ(x), d [µ(x)y(x)]. dx

µ(x)y 0 (x) + µ(x)p(x)y(x) =

For this purpose, the function µ(x) must has the property µ0 (x) = p(x)µ(x),

(2.1)

and µ(x) 6= 0 for all x. By solving the linear homogeneous equation (2.1), one obtain R

µ(x) = e

p(x)dx

.

(2.2)

With this function, which is called an integrating factor, our equation is reduced to d [µ(x)y(x)] = µ(x)q(x), dx

(2.3)

Integrating both sides, we get Z

µ(x)y =

µ(x)q(x)dx + C

with C an arbitrary constant. Solving for y, we get y=

1 µ(x)

Z

µ(x)q(x)dx +

C = yP (x) + yH (x) µ(x)

(2.4)

9

FIRST ORDER DIFFERENTIAL EQUATIONS

as the general solution for the general linear first order ODE y 0 + p(x)y = q(x). In solution (2.4), the first part, yP (x), is a particular solution of the inhomogeneous equation, while the second part, yH (x), is the general solution of the associate homogeneous solution. Note that for any pair of scalars a, b with a in (I), there is a unique scalar C such that y(a) = b. Geometrically, this means that the solution curves y = φ(x) are a family of non-intersecting curves which fill the region I × R. Example 1: y 0 + xy = x. This is a linear first order ODE in standard form with p(x) = q(x) = x. The integrating factor is R

xdx

µ(x) = e

2 /2

= ex

.

Hence, after multiplying both sides of our differential equation, we get d x2 /2 2 (e y) = xex /2 dx which, after integrating both sides, yields 2 /2

ex

Z

2 /2

xex

y=

dx + C = ex

2 /2

+ C.

2

Hence the general solution is y = 1+Ce−x /2 . The solution satisfying the initial condition y(0) = 1 is y = 1 and the solution satisfying y(0) = a 2 is y = 1 + (a − 1)e−x /2 . Example 2: xy 0 − 2y = x3 sin x, (x > 0). We bring this linear first order equation to standard form by dividing by x. We get y0 + The integrating factor is µ(x) = e

R

−2 y = x2 sin x. x

−2dx/x

= e−2 ln x = 1/x2 .

After multiplying our DE in standard form by 1/x2 and simplifying, we get d (y/x2 ) = sin x dx from which y/x2 = − cos x + C and y = −x2 cos x + Cx2 . Note that the later are solutions to the DE xy 0 − 2y = x3 sin x and that they all satisfy the initial condition y(0) = 0. This non-uniqueness is due to the fact that x = 0 is a singular point of the DE.

PART (II): SEPARABLE EQUATIONS — NONLINEAR EQUATIONS (1)

2.

Separable Equations.

The first order ODE y 0 = f (x, y) is said to be separable if f (x, y) can be expressed as a product of a function of x times a function of y. The DE then has the form y 0 = g(x)h(y) and, dividing both sides by h(y), it becomes y0 = g(x). h(y) Of course this is not valid for those solutions y = y(x) at the points where φ(x) = 0. Assuming the continuity of g and h, we can integrate both sides of the equation to get Z

y 0 (x) dx = h[y(x)]

Assume that

Z

g(x)dx + C. Z

H(y) =

dy , h(y)

By chain rule, we have d 1 H[y(x)] = H 0 (y)y 0 (x) = y 0 (x), dx h[y(x)] hence

Z

H[y(x)] = Therefore,

Z

y 0 (x) dx = h[y(x)]

dy = H(y) = h(y)

11

Z

g(x)dx + C.

Z

g(x)dx + C,

12

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

gives the implicit form of the solution. It determines the value of y implicitly in terms of x. Example 1: y 0 = x−5 . y2 To solve it using the above method we multiply both sides of the equation by y 2 to get y 2 y 0 = (x − 5). Integrating both sides we get y 3 /3 = x2 /2 − 5x + C. Hence, h

y = 3x2 /2 − 15x + C1

i1/3

.

y−1 Example 2: y 0 = x+3 (x > −3). By inspection, y = 1 is a solution. Dividing both sides of the given DE by y − 1 we get

1 y0 = . y−1 x+3 This will be possible for those x where y(x) 6= 1. Integrating both sides we get Z Z dx y0 dx = + C1 , y−1 x+3 from which we get ln |y − 1| = ln(x + 3) + C1 . Thus |y − 1| = eC1 (x + 3) from which y − 1 = ±eC1 (x + 3). If we let C = ±eC1 , we get y = 1 + C(x + 3) which is a family of lines passing through (−3, 1); for any (a, b) with b 6= 0 there is only one member of this family which passes through (a, b). Since y = 1 was found to be a solution by inspection the general solution is y = 1 + C(x + 3), where C can be any scalar. y cos x Example 3: y 0 = 1+2y Transforming in the standard form then 2. integrating both sides we get

Z

(1 + 2y 2 ) dy = y

Z

cos x dx + C,

from which we get a family of the solutions: ln |y| + y 2 = sin x + C,

FIRST ORDER DIFFERENTIAL EQUATIONS

13

where C is an arbitrary constant. However, this is not the general solution of the equation, as it does not contains, for instance, the solution: y = 0. With I.C.: y(0)=1, we get C = 1, hence, the solution: ln |y| + y 2 = sin x + 1.

3.

Logistic Equation y 0 = ay(b − y),

where a, b > 0 are fixed constants. This equation arises in the study of the growth of certain populations. Since the right-hand side of the equation is zero for y = 0 and y = b, the given DE has y = 0 and y = b as solutions. More generally, if y 0 = f (t, y) and f (t, c) = 0 for all t in some interval (I), the constant function y = c on (I) is a solution of y 0 = f (t, y) since y 0 = 0 for a constant function y. To solve the logistic equation, we write it in the form y0 = a. y(b − y) Integrating both sides with respect to t we get Z

y 0 dt = at + C y(b − y)

which can, since y 0 dt = dy, be written as Z

dy = at + C. y(b − y)

Since, by partial fractions, 1 1 1 1 = ( + ) y(b − y) b y b−y we obtain

1 (ln |y| − ln |b − y|) = at + C. b Multiplying both sides by b and exponentiating both sides to the base e, we get |y| = ebC eabt = C1 eabt , |b − y| where the arbitrary constant C1 = ±ebC can be determined by the initial condition (IC): y(0) = y0 as C1 =

|y0 | . |b − y0 |

14

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

Two cases need to be discussed separately. y0 Case (I), y0 < b: one has C1 = | b−y |= 0

|y| = |b − y|

µ

y0 b−y0

> 0. So that,

¶

y0 eabt > 0, b − y0

(t ∈ (I)).

From the above we derive y/(b − y) = C1 eabt , and y = (b − y)C1 eabt . This gives ³ ´ y0 b b−y eabt bC1 eabt 0 ³ ´ y= = . 1 + C1 eabt 1 + y0 eabt b−y0

It shows that if y0 = 0, one has the solution y(t) = 0. However, if 0 < y0 < b, one has the solution 0 < y(t) < b, and as t → ∞, y(t) → b. y0 y0 Case (II), y0 > b: one has C1 = | b−y | = − b−y > 0. So that, 0 0

¯ µ ¯ ¶ ¯ y ¯ y0 ¯= ¯ eabt > 0, (t ∈ (I)). ¯b − y ¯ y0 − b ´ ³

From the above we derive y/(y − b) = ³

b)

y0 y0 −b

´

y0 y0 −b

eabt , and y = (y −

eabt . This gives ³

y=³

b

y0 y0 −b

y0 y0 −b

´

´

eabt

eabt − 1

.

It shows that if y0 > b, one has the solution y(t) > b, and as t → ∞, y(t) → b. It is derived that y(t) = 0 is an unstable equilibrium state of the system; y(t) = b is a stable equilibrium state of the system.

4.

Fundamental Existence and Uniqueness Theorem

If the function f (x, y) together with its partial derivative with respect to y are continuous on the rectangle R : |x − x0 | ≤ a, |y − y0 | ≤ b there is a unique solution to the initial value problem y 0 = f (x, y),

y(x0 ) = y0

FIRST ORDER DIFFERENTIAL EQUATIONS

15

defined on the interval |x − x0 | < h where h = min(a, b/M ),

M = max |f (x, y)|, (x, y) ∈ R.

Note that this theorem indicates that a solution may not be defined for all x in the interval |x − x0 | ≤ a. For example, the function y=

bCeabx 1 + Ceabx

is solution to y 0 = ay(b − y) but not defined when 1 + Ceabx = 0 even though f (x, y) = ay(b − y satisfies the conditions of the theorem for all x, y. The next example show why the condition on the partial derivative in the above theorem is necessary. Consider the differential equation y 0 = y 1/3 . Again y = 0 is a solution. Separating variables and integrating, we get Z

dy = x + C1 y 1/3

which yields y 2/3 = 2x/3 + C and hence y = ±(2x/3 + C)3/2 . Taking C = 0, we get the solution y = (2x/3)3/2 , (x ≥ 0) which along with the solution y = 0 satisfies y(0) = 0. So the initial value problem y 0 = y 1/3 , y(0) = 0 does not have a unique solution. The reason this 2/3 is not continuous when is so is due to the fact that ∂f ∂y (x, y) = 1/3y y = 0. Many differential equations become linear or separable after a change of variable. We now give two examples of this.

5.

Bernoulli Equation: y 0 = p(x)y + q(x)y n

(n 6= 1).

Note that y = 0 is a solution. To solve this equation, we set u = y α , where α is to be determined. Then, we have u0 = αy α−1 y 0 , hence, our differential equation becomes u0 /α = p(x)u + q(x)y α+n−1 .

(2.5)

Now set α = 1 − n. Thus, (2.5) is reduced to u0 /α = p(x)u + q(x),

(2.6)

which is linear. We know how to solve this for u from which we get solve u = y 1−n to get y.

16

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

6.

Homogeneous Equation: y 0 = F (y/x).

To solve this we let u = y/x so that y = xu and y 0 = u+xu0 . Substituting for y, y 0 in our DE gives u + xu0 = F (u) which is a separable equation. Solving this for u gives y via y = xu. Note that u = a is a solution of xu0 = F (u) − u whenever F (a) = a and that this gives y = ax as a solution of y 0 = f (y/x). Example. y 0 = (x − y)/x + y. This is a homogeneous equation since x−y 1 − y/x = . x+y 1 + y/x Setting u = y/x, our DE becomes xu0 + u =

1−u 1+u

so that

1 − 2u − u2 1−u −u= . 1+u 1+u √ Note that the right-hand side is zero if u = −1± 2. Separating variables and integrating with respect to x, we get xu0 =

Z

(1 + u)du = ln |x| + C1 1 − 2u − u2

which in turn gives (−1/2) ln |1 − 2u − u2 | = ln |x| + C1 . Exponentiating, we get 1 = eC1 |x|. |1 − 2u − u2 |

p

Squaring both sides and taking reciprocals, we get u2 + 2u − 1 = C/x2 with C = ±1/e2C1 . This equation can be solved for u using the quadratic formula. If x0 , y0 are given with x0 6= 0 and u0 = y0 /x0 6= −1 there is, by the fundamental, existence and uniqueness theorem,a unique solution with u(x0 ) = y0 . For example, if x0 = 1, y0 = 2, we have C = 7 and hence u2 + 2u − 1 = 7/x2

17

FIRST ORDER DIFFERENTIAL EQUATIONS

Solving for u, we get

q

u = −1 +

2 + 7/x2

where the positive sign in the quadratic formula was chosen to make u = 2, x = 1 a solution. Hence q

y = −x + x 2 + 7/x2 = −x +

p

2x2 + 7

is the solution to the initial value problem y0 =

x−y , x+y

y(1) = 2

for x > 0 and one can easily check that it is a solution for all x. Moreover, using the fundamental uniqueness, it can be shown that it is the only solution defined for all x.

PART (III): EXACT EQUATION AND INTEGRATING FACTOR — NONLINEAR EQUATIONS (2)

7.

Exact Equations.

By a region of the xy-plane we mean a connected open subset of the plane. The differential equation M (x, y) + N (x, y)

dy =0 dx

is said to be exact on a region (R) if there is a function F (x, y) defined on (R) such that ∂F = M (x, y); ∂x

∂F = N (x, y) ∂y

In this case, if M, N are continuously differentiable on (R) we have ∂M ∂N = . ∂y ∂x

(2.7)

Conversely, it can be shown that condition (2.7) is also sufficient for the exactness of the given DE on (R) providing that (R) is simply connected, .i.e., has no “holes”. The exact equations are solvable. In fact, suppose y(x) is its solution. Then one can write: M [x, y(x)] + N [x, y(x)]

∂F ∂F dy d dy = + = F [x, y(x)] = 0. dx ∂x ∂y dx dx

It follows that F [x, y(x)] = C,

19

20

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

where C is an arbitrary constant. This is an implicit form of the solution y(x). Hence, the function F (x, y), if it is found, will give a family of the solutions of the given DE. The curves F (x, y) = C are called integral curves of the given DE. dy Example 1. 2x2 y dx + 2xy 2 + 1 = 0. Here M = 2xy 2 + 1, N = 2x2 y and R = R2 , the whole xy-plane. The equation is exact on R2 since R2 is simply connected and

∂M ∂N = 4xy = . ∂y ∂x To find F we have to solve the partial differential equations ∂F = 2xy 2 + 1, ∂x

∂F = 2x2 y. ∂y

If we integrate the first equation with respect to x holding y fixed, we get F (x, y) = x2 y 2 + x + φ(y). Differentiating this equation with respect to y gives ∂F = 2x2 y + φ0 (y) = 2x2 y ∂y using the second equation. Hence φ0 (y) = 0 and φ(y) is a constant function. The solutions of our DE in implicit form is x2 y 2 + x = C. Example 2. We have already solved the homogeneous DE dy x−y = . dx x+y This equation can be written in the form y − x + (x + y)

dy =0 dx

which is an exact equation. In this case, the solution in implicit form is x(y − x) + y(x + y) = C, i.e., y 2 + 2xy − x2 = C.

8.

Theorem. If F (x, y) is homogeneous of degree n then x

∂F ∂F +y = nF (x, y). ∂x ∂y

21

FIRST ORDER DIFFERENTIAL EQUATIONS

Proof. The function F is homogeneous of degree n if F (tx, ty) = tn F (x, y). Differentiating this with respect to t and setting t = 1 yields the result. QED

9.

Integrating Factors.

If the differential equation M + N y 0 = 0 is not exact it can sometimes be made exact by multiplying it by a continuously differentiable function µ(x, y). Such a function is called an integrating factor. An integrating ∂µN factor µ satisfies the PDE ∂µM ∂y = ∂x which can be written in the form µ

∂M ∂N − ∂y ∂x

¶

µ=N

∂µ ∂µ −M . ∂x ∂y

This equation can be simplified in special cases, two of which we treat next. µ is a function of x only.

This happens if and only if ∂M ∂y

−

∂N ∂x

N

= p(x)

is a function of x only in which case µ0 = p(x)µ. µ is a function of y only.

This happens if and only if ∂M ∂y

−

∂N ∂x

M

= q(y)

is a function of y only in which case µ0 = −q(y)µ. µ = P (x)Q(y) .

This happens if and only if ∂M ∂N − = p(x)N − q(y)M, ∂y ∂x

where p(x) =

P 0 (x) , P (x)

q(y) =

(2.8)

Q0 (y) . Q(y)

If the system really permits the functions p(x), q(y), such that (2.8) hold, then we can derive R

P (x) = ±e

p(x)dx

R

;

Q(y) = ±e

q(y)dy

.

22

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

Example 1. 2x2 + y + (x2 y − x)y 0 = 0. Here ∂M ∂y

−

∂N ∂x

N

=

2 − 2xy −2 = x2 y − x x

so that there is an integrating factor µ which is a function of x only which satisfies µ0 = −2µ/x. Hence µ = 1/x2 is an integrating factor and 2 + y/x2 + (y − 1/x)y 0 = 0 is an exact equation whose general solution is 2x − y/x + y 2 /2 = C or 2x2 − y + xy 2 /2 = Cx. Example 2. y + (2x − yey )y 0 = 0. Here ∂M ∂y

− M

∂N ∂x

=

−1 y

so that there is an integrating factor which is a function of y only which satisfies µ0 = 1/y. Hence y is an integrating factor and y 2 + (2xy − y 2 ey )y 0 = 0 is an exact equation with general solution xy 2 + (−y 2 + 2y − 2)ey = C. A word of caution is in order here. The solutions of the exact DE obtained by multiplying by the integrating factor may have solutions which are not solutions of the original DE. This is due to the fact that µ may be zero and one will have to possibly exclude those solutions where µ vanishes. However, this is not the case for the above Example 2.

PART (IV): CHANGE OF VARIABLES — NONLINEAR EQUATIONS (3)

10.

Change of Variables.

Sometimes it is possible by means of a change of variable to transform a DE into one of the known types. For example, homogeneous equations can be transformed into separable equations and Bernoulli equations can be transformed into linear equations. The same idea can be applied to some other types of equations, as described as follows.

10.1

y 0 = f (ax + by), b 6= 0

Here, if we make the substitution u = ax+by the differential equation becomes du = bf (u) + a dx which is separable. √ √ Example 1. The DE y 0 = 1 + y − x becomes u0 = u after the change of variable u = y − x.

10.2

dy dx

=

a1 x + b1 y + c1 a2 x + b2 y + c2

Here, we assume that a1 x+b1 y +c1 = 0, a2 x+b2 y +c2 = 0 are distinct lines meeting in the point (x0 , y0 ). The above DE can be written in the form a1 (x − x0 ) + b1 (y − y0 ) dy = dx a2 (x − x0 ) + b2 (y − y0 ) which yields the DE a1 X + b1 Y dY = dX a2 X + b2 Y

23

24

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

after the change of variables X = x − x0 , Y = y − y0 .

10.3

Riccatti equation: y 0 = p(x)y + q(x)y 2 + r(x)

Suppose that u = u(x) is a solution of this DE and make the change of variables y = u + 1/v. Then y 0 = u0 − v 0 /v 2 and the DE becomes u0 − v 0 /v 2 = p(x)(u + 1/v) + q(x)(u2 + 2u/v + 1/v 2 ) + r(x) = p(x)u + q(x)u2 + r(x) + (p(x) + 2uq(x))/v + q(x)/v 2 from which we get v 0 + (p(x) + 2uq(x))v = −q(x), a linear equation. Example 2. y 0 = 1 + x2 − y 2 has the solution y = x and the change of variable y = x + 1/v transforms the equation into v 0 + 2xv = 1.

PART (V): SOME APPLICATIONS

We now give a few applications of differential equations.

11.

Orthogonal Trajectories.

An important application of first order DE’s is to the computation of the orthogonal trajectories of a family of curves f (x, y, C) = 0. An orthogonal trajectory of this family is a curve that, at each point of intersection with a member of the given family, intersects that member orthogonally. To find the orthogonal trajectories, we may derive the ODE, whose solutions are described by these trajectories. For this purpose, we are going first to derive the ODE, whose solutions have the implicit form, f (x, y, C) = 0. In doing so, we differentiate f (x, y, C) = 0 implicitly with respect to x we get ∂f ∂f 0 + y =0 ∂x ∂y from which we get y0 = −

fx (x, y, C) . fy (x, y, C)

Now we solve for C = C(x, y) from the equation f (x, y, C) = 0, which specifies the curve passing through the point (x, y). We substitute C(x, y) in the above formula for y 0 . This gives the equation: h

y 0 = g(x, y) = −

i

fx x, y, C(x, y) h

i.

fy x, y, C(x, y)

Note that y 0 (x) yields the slope of the tangent line at the point (x, y) of a curve of the given family passing through (x, y). The slope of the

25

26

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

orthogonal trajectory at the passing point (x, y) must be y 0 (x) = −

1 . g(x, y)

Therefore, the ODE governing the orthogonal trajectories is derived as h

y0 =

i

fy x, y, C(x, y) h

i.

fx x, y, C(x, y)

Example 3. Let us find the orthogonal trajectories of the family x2 + y 2 = Cx, the family of circles with center on the x-axis and passing through the origin. Here x2 + y 2 x 0 from which, we derive the ODE: y = g(x, y) = (y 2 − x2 )/2xy. Then the ODE governing the orthogonal trajectories can be written as 1 y0 = − , g(x, y) or, y 0 = 2xy/(x2 − y 2 ). 2x + 2yy 0 = C =

The above can be re-written in the form: 2xy + (y 2 − x2 )y 0 = 0. If we let M = 2xy, N = y 2 − x2 we have ∂M ∂y

−

∂N ∂x

M

=

4x 2 = 2xy y

so that we have an integrating factor µ which is a function of y. We have µ0 = −2µ/y from which µ = 1/y 2 . Multiplying the DE for the orthogonal trajectories by 1/y 2 we get Ã

2x x2 + 1− 2 y y

!

y 0 = 0.

∂F 2 2 2 Solving ∂F ∂x = 2x/y, ∂y = 1 − x /y for F yields F (x, y) = x /y + y from which the orthogonal trajectories are x2 /y + y = C, i.e., x2 + y 2 = Cy. This is the family of circles with center on the y-axis and passing through the origin. Note that the line y = 0 is also an orthogonal trajectory that was not found by the above procedure. This is due to the fact that the integrating factor was 1/y 2 which is not defined if y = 0 so we had to work in a region which does not cut the x-axis, e.g., y > 0 or y < 0.

FIRST ORDER DIFFERENTIAL EQUATIONS

12.

27

Falling Bodies with Air Resistance

Let x be the height at time t of a body of mass m falling under the influence of gravity. If g is the force of gravity and b v is the force on the body due to air resistance, Newton’s Second Law of Motion gives the DE dv m = mg − bv dt where v = dx dt . This DE has the general solution v(t) =

mg + Be−bt/m . b

The limit of v(t) as t → ∞ is mg/b, the terminal velocity of the falling body. Integrating once more, we get x(t) = A +

13.

mg t mB −bt/m − e . b b

Mixing Problems

Suppose that a tank is being filled with brine at the rate of a units of volume per second and at the same time b units of volume per second are pumped out. If the concentration of the brine coming in is c units of weight per unit of volume. If at time t = t0 the volume of brine in the tank is V0 and contains x0 units of weight of salt, what is the quantity of salt in the tank at any time t, assuming that the tank is well mixed? If x is the quantity of salt at any time t, we have ac units of weight of salt coming in per second and b

bx x(t) = V (t) V0 + (a − b)(t − t0 )

units of weight of salt going out. Hence dx bx = ac − , dt V0 + (a − b)(t − t0 ) a linear equation. If a = b it has the solution x(t) = cV0 + (x0 − cV0 )e−a(t−t0 )/V0 . As a numerical example, suppose a = b = 1 liter/min, c = 1 grams/liter, V0 = 1000 liters, x0 = 0 and t0 = 0. Then x(t) = 1000(1 − e−.001t ) is the quantity of salt in the tank at any time t. Suppose that after 100 minutes the tank springs a leak letting out an additional liter of brine

28

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

per minute. To find out how much salt is in the tank 12 hours after the leak begins we use the DE dx 2x 2 =1− =1− x. dt 1000 − (t − 100) 1100 − t This equation has the general solution x(t) = (1100 − t)−1 + C(1100 − t)2 . Using x(100) = 1000(1 − e−.1 ) = 95.16, we find C = −9.048 × 10−4 and x(820) = 177.1. When t = 1100 the tank is empty and the differential equation is no a valid description of the physical process. The concentration at time 100 < t < 1100 is x(t) = 1 + C(1100 − t) 1100 − t which converges to 1 as t tends to 1100.

14.

Heating and Cooling Problems

Newton’s Law of Cooling states that the rate of change of the temperature of a cooling body is proportional to the difference between its temperature T and the temperature of its surrounding medium. Assuming the surroundings maintain a constant temperature Ts , we obtain the differential equation dT = −k(T − Ts ), dt where k is a constant. This is a linear DE with solution T = Ts + Ce−kt . If T (0) = T0 then C = T0 − Ts and T = Ts + (T0 − Ts )e−kt . As an example consider the problem of determining the time of death of a healthy person who died in his home some time before noon when his body was 70 degrees. If his body cooled another 5 degrees in 2 hours when did he die, assuming that the room was a constant 60 degrees. Taking noon as t = 0 we have T0 = 70. Since Ts = 60, we get 65 − 60 = 10e−2k from which k = ln(2)/2. To determine the time of death we use the equation 98.6 − 60 = 10e−kt which gives t = − ln(3.86)/k = −2 ln(3.86)/ ln(2) = −3.90. Hence the time of death was 8 : 06 AM.

FIRST ORDER DIFFERENTIAL EQUATIONS

15.

29

Radioactive Decay

A radioactive substance decays at a rate proportional to the amount of substance present. If x is the amount at time t we have dx = −kx, dt where k is a constant. The solution of the DE is x = x(0)e−kt . If c is the half-life of the substance we have by definition x(0)/2 = x(0)e−kc which gives k = ln(2)/c.

PART (VI)*: GEOMETRICAL APPROACHES — NONLINEAR EQUATIONS (4)

16. 16.1

Definitions and Basic Concepts Directional Field

A plot of short line segments drawn at various points in the (x, y) plane showing the slope of the solution curve there is called direction field for the DE.

16.2

Integral Curves

The family of curves in the (x, y) plane, that represent all solutions of DE is called the integral curves.

16.3

Autonomous Systems

The first order DE dy/dx = f (y) is called autonomous, since the independent variable does not appear explicitly. The isoclines are made up of horizonal lines y = m, along which the slope of directional fields is the constant, y 0 = f (m).

16.4

Equilibrium Points

The DE has the constant solution y = y0 , if and only if f (y0 ) = 0. These values of y0 are the equilibrium points or stationary points of the DE. y = y0 is called a source if f (y) changes sign from - to + as y increases from just below y = y0 to just above y = y0 and is called a sink if f (y) changes sign from + to - as y increases from just below y = y0 to just above y = y0 ; it is called a node if there is no change in

31

32

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

sign. Solutions y(t) of the DE appear to be attracted by the line y = y0 , if y0 is a sink and move away from the line y = y0 , if y0 is a source.

17.

Phase Line Analysis

The y-axis on which is plotted the equilibrium points of the DE with arrows between these points to indicate when the solution y is increasing or decreasing is called the phase line of the DE. The autonomous DE dy/dx = 2y − y 2 has 0 and 1 as equilibrium points. The point y = 0 is a source and y = 2 is a sink (see Fig.2.1). This DE is a logistic model for a population having 2 as the size of a stable population. The equation dy/dx = −y(2 − y)(3 − y) has three equilibrium states: y = 0, 2, 3. Among them, y = 0, 3 are the sink, while y = 2 is the source (see Fig.2.2). The equation dy/dx = −y(2 − y)2 has two equilibrium states: y = 0, 2. The point y = 0 is a sink, while y = 2 is a node (see Fig.2.3). The sink is stable, source is unstable, whereas the node is semi-stable. The node point of the equation y = f (y) can either disappear, or split into one sink and one source, when the equation is perturbed with a small amount ε and becomes: y = f (y) + ε.

18.

Bifurcation Diagram

Some dynamical system contains a parameter Λ, such as y 0 = f (y, Λ). Then the characteristics of its equilibrium states, such as their number and nature, depends on the value of Λ. Some times, through a special value of Λ = Λ∗ , these characteristics of equilibrium states may change. This Λ = Λ∗ is called the bifurcation point. Example 1. For the logistic population growth model, if the population is reduced at a constant rate s > 0, the DE becomes dy/dx = 2y − y 2 − s which has a source at the larger of the two roots of the equation y 2 − 2y + s = 0

33

FIRST ORDER DIFFERENTIAL EQUATIONS

y 0 = f (y)

y=0

y=2 y

Figure 2.1. Sketch of the phase line for the equation dy/dx = 2y − y 2 , in which y = 0 is a source, y = 2 is a sink.

y 0 = f (y)

y=0

y=2

y=3

y

Figure 2.2. Sketch of the phase line for the equation dy/dx = −y(2 − y)(3 − y), in which y = 0, 3 is a sink, y = 2 is a source.

for s < 2. If s > 2 there is no equilibrium point and the population dies out as y is always decreasing. The point s=2 is called a bifurcation point of the DE. Example 2. Chemical Reaction Model. One has the DE ·

dy/dx = −ay y 2 − where

¸

R − Rc , a

34

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

y 0 = f (y)

y=0

y=2 y

Figure 2.3. Sketch of the phase line for the equation dy/dx = −y(2 − y)2 , in which the point y = 0 is a sink, while y = 2 is a node.

y is the concentration of species A; R is the concentration of some chemical element, and (a, Rc ) are constants (fixed). It is derived that If R < Rc , the system has one equilibrium state y = 0, which is stable; If R > Rc , the system has q three equilibrium states: y = 0, which is c now unstable, and y = ± R−R a , which are stable. For this system, R = Rc is the bifurcation point.

35

FIRST ORDER DIFFERENTIAL EQUATIONS

y 0 = f (y, s)

s = 1.5 y y0 (s) s = 1.0 s = 1.5 s = 2

s

Figure 2.4. Sketch of the bifurcation diagram of the equation dy/dx = y(2 − y) − s, in which the point s = 2 is the bifurcation point.

36

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

y 0 = f (y, R)

y y0 (R)

R < Rc R = Rc

R > Rc

R

Figure of the bifurcation diagram of the equation dy/dx ¤ £ 2.5. Sketch c , in which the point R = Rc is the bifurcation point. −ay y 2 − R−R a

=

PART (VII): NUMERICAL APPROACH AND APPROXIMATIONS

19.

Euler’s Method

In this section we discuss methods for obtaining a numerical solution of the initial value problem y 0 = f (x, y),

y(x0 ) = y0

at equally spaced points x0 , x1 , x2 , . . . , xN = p, . . . where xn+1 − xn = h > 0 is called the step size. In general, the smaller the value of the better the approximations will be but the number of steps required will be larger. We begin by integrating y 0 = f (x, y) between xn and xn+1 . If y(x) = φ(x), this gives φ(xn+1 ) = φ(xn ) +

Z xn+1 xn

f (t, φ(t))dt.

As a first estimate of the integrand we use the value of f (t, φ(t)) at the lower limit xn , namely f (xn , φ(xn )). Now, assuming that we have already found an estimate yn for φ(xn ), we get the estimate yn+1 = yn + hf (xn , yn ) for φ(xn+1 ). It can be shown that |yn − φ(xn )| ≤ Ch, where C is a constant which depends on p.

37

38

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

20.

Improved Euler’s Method

The Euler method can be improved if we use the trapezoidal rule for estimating the above integral. Namely, Z b a

1 F (x)dx = (F (a) + F (b))(b − a). 2

This leads to the estimate yn+1 = yn +

h (f (xn , yn ) + f (xn+1 , yn+1 )). 2

If we now use the Euler approximation yn+1 to compute f (xn+1 , yn+1 ), we get yn+1 = yn +

h (f (xn , yn ) + f (xn + h, yn + hf (xn , yn )). 2

This is known as the improved Euler method. It can be shown that |yn − φ(xn )| ≤ Ch2 . In general, if yn is an approximation for φ(xn ) such that |yn − φ(xn )| ≤ Chp , we say that the approximation is of order p. Thus the Euler method is first order and the improved Euler is second order.

21.

Higher Order Methods

On can obtain higher order approximations by using better approximations for F (t) = f (t, φ(t)) on the interval [xn , xn+1 ]. For example, the Taylor series approximation F (t) = F (xn )+F 0 (xn )(t−xn )+F 00 (xn )(t−xn )2 /2+· · ·+F (p−1) (xn )(t−xn )p−1 /(p−1)! yields the approximation yn+1 = yn + hf1 (xx , yn ) +

h2 hp f2 (xn , yn ) + · · · + fp−1 (xn , yn ), 2 p!

where ·

fk (xn , yn ) = F

(k−1)

∂ ∂ (xn ) = + f (x, y) ∂x ∂y

¸(k−1)

f (xn , yn ).

It can be show that this approximation is of order p. However it is computationally intensive as one has to compute higher derivatives.

39

FIRST ORDER DIFFERENTIAL EQUATIONS

In the case p = 2 this formula was simplified by Runge and Kutta to give the second order midpoint approximation ·

¸

yn+1 = yn + hf xn +

h h , yn + f (xn , yn ) . 2 2

In the case p = 4 they obtained the 4-th order approximation 1 yn+1 = yn + (k1 + 2k2 + 2k3 + k4 ), 6 where k1 k2 k3 k4

= hf (xn , yn ), = hf (xn + h2 , yn + k21 ), = hf (xn + h2 , yn + k22 ), = hf (xn + h, yn + k3 ).

(2.9)

Computationally, it is much simpler than the 4-th order Taylor series approximation from which it is derived.

4() Picard Iteration We assume that f (x, y) and

∂f ∂y

are continuous on the rectangle

R : |x − x0 | ≤ a, |y − y0 | ≤ b Then |f (x, y)| ≤ M , | ∂f ∂y (x, y)| ≤ L on R. The initial value problem 0 y = f (x, y), y(x0 ) = y0 is equivalent to the integral equation y = y0 +

Z x x0

f (t, y(t))dt.

Let the righthand side of the above equation be denoted by T (y). Then our problem is to find a solution to y = T (y) which is a fixed point problem. To solve this problem we take as an initial approximation to y the constant function y0 (x) = y0 and consider the iterations yn = T n (y0 ). The function yn is called the n-th Picard iteration of y0 . For example, for the initial value problem y 0 = x + y 2 , y(0) = 1 we have y1 (x) = 1 + y2 (x) = 1+

Z x 0

Z x 0

(t + 1)dt = 1 + x + x2 /2

(t+(1+t+t2 /2)2 )dt = 1+x+3x2 /2+2x3 /3+x4 /4+x5 /20.

Contrary to the power series approximations we can determine just how good the Picard iterations approximate y. In fact, we will see that the Picard iterations converge to a solution of our initial value problem. More precisely we have the following result:

40

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

4.1 Theorem of Existence and Uniqueness of Solution for IVP The Picard iterations yn = T n (y0 ) converge to a solution y of y 0 = f (x, y), y(x0 ) = y0 on the interval |x−x0 | ≤ h = min(a, b/M ). Moreover |y(x) − yn (x)| ≤ (M/L)ehL (Lh)n+1 /(n + 1)! for |x − x0 | ≤ h and the solution y is unique on this interval. Proof. We have |y1 − y0 | = |

Z x x0

f (t, y0 )| ≤ M |x − x0 |

since |f (x, y)| ≤ M on R. Now |y1 − y0 | ≤ b if |x − x0 | ≤ h. So (x, y1 (x)) is in R if |x−x0 | ≤ h. Similarly, one can show inductively that (x, yn (x)) is in R if |x − x0 | ≤ h. Using the fact that, by the mean value theorem for derivatives, |f (x, z) − f (x, w| ≤ L|z − w| for all (x, w), (x, z) in R, we obtain Z x

|y2 − y1 | = |

x0

Z x

|y3 − y2 | = |

x0

(f (t, y1 ) − f (t, y0 )| ≤ M L|x − x0 |2 /2,

(f (t, y2 ) − f (t, y1 )| ≤ M L2 |x − x0 |3 /6

and by induction |yn − yn−1 | ≤ M Ln−1 |x − x0 |n /n!. Since the series − yn−1 | is bounded above term by term by the convergent series 1 |ynP n (M/L) ∞ 1 (L|x − x0 |) /n!, its n-th partial sum yn − y0 converges, which gives the convergence of yn to a function y. Now since P∞

y = y0 + (y1 − y0 ) + · · · + (yn − yn−1 ) +

∞ X

(yi − yi−1 )

i=n+1

we obtain |y − yn | ≤

∞ X

(M/L)(L(|x − x0 |)i /i! ≤ (M/L)

i=n+1

(Lh)n+1 hL e . (n + 1)!

For the uniqueness, suppose T (z) = z with (x, z(x) in R for |x − x0 | ≤ h. Then Z x y(x) − z(x) = (f (t, y(x)) − f (t, z(x))dt. x0

If |y(x) − z(x)| ≤ A for x − x0 | ≤ h we then obtain as above |y(x) − z(x)| ≤ AL|x − x0 |.

FIRST ORDER DIFFERENTIAL EQUATIONS

41

Now using this estimate, repeat the above to get |y(x) − z(x)| ≤ AL2 |x − x0 |2 /2. Using induction we get that |y(x) − z(x)| ≤ ALn |x − x0 |n /n! which converges to zero for all x. Hence y = z.

QED

The key ingredient in the proof is the Lipschitz Condition |f (x, y) − f (x, z)| ≤ L|y − z|. If f (x, y) is continuous for |x − x0 | ≤ a and all y and satisfies the above Lipschitz condition in this strip the above proof gives the existence and uniqueness of the solution to the initial value problem y 0 = f (x, y), y(x0 ) = y0 on the interval |x − x0 | ≤ a.

Chapter 3 N-TH ORDER DIFFERENTIAL EQUATIONS

43

PART (I): THE FUNDAMENTAL EXISTENCE AND UNIQUENESS THEOREM

In this lecture we will state and sketch the proof of the fundamental existence and uniqueness theorem for the n-th order DE y (n) = f (x, y, y 0 , . . . , y (n−1) ). The starting point is to convert this DE into a system of first order DE’. Let y1 = y, y2 = y 0 , . . . y (n−1) = yn . Then the above DE is equivalent to the system dy1 dx dy2 dx

= y2 = y3 .. .

dyn dx

= f (x, y1 , y2 , . . . , yn ).

(3.1)

More generally let us consider the system dy1 dx dy2 dx

= f1 (x, y1 , y2 , . . . , yn ) = f2 (x, y1 , y2 , . . . , yn ) .. .

dyn dx

= fn (x, y1 , y2 , . . . , yn ). n

(3.2) o

If we let Y = (y1 , y2 , . . . , yn ), F (x, Y ) = f1 (x, Y ), f2 (x, Y ), . . . , fn (x, Y ) and

dY dx

dyn 1 dy2 = ( dy dx , dx , . . . , dx ), the system becomes

dY = F (x, Y ). dx

45

46

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

1.

Theorem of Existence and Uniqueness (I)

∂fi are continuous on the n + 1-dimensional If fi (x, y1 , . . . , yn ) and ∂y j box R : |x − x0 | < a, |yi − ci | < b, (1 ≤ i ≤ n)

for 1 ≤ i, j ≤ n with |fi (x, y)| ≤ M and ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ∂fi ¯ ¯ ∂fi ¯ ¯+¯ ¯ + . . . ¯ ∂fi ¯ < L ¯ ¯ ∂y ¯ ¯ ∂y ¯ ¯ ∂y ¯ 1

2

n

on R for all i, the initial value problem dY = F (x, Y ), dx

Y (x0 ) = (c1 , c2 , . . . , cn )

has a unique solution on the interval |x − x0 | ≤ h = min(a, b/M ). The proof is exactly the same as for the proof for n = 1 if we use the following Lemma in place of the mean value theorem.

1.1

Lemma

If f (x1 , x2 , . . . xn ) and its partial derivatives are continuous on an ndimensional box R, then for any a, b ∈ R we have ¯ ¯¶ ¯ µ¯ ¯ ¯ ¯ ∂f ¯ ∂f ¯ ¯ ¯ |f (a) − f (b)| ≤ ¯ (c)¯ + · · · + ¯ (c)¯¯ |a − b| ∂x ∂x 1

n

where c is a point on the line between a and b and |(x1 , . . . , xn )| = max(|x1 |, . . . , |xn |). The lemma is proved by applying the mean value theorem to the function G(t) = f (ta + (1 − t)b). This gives G(1) − G(0) = G0 (c) for some c between 0 and 1. The lemma follows from the fact that G0 (x) =

∂f ∂f (a1 − b1 ) + · · · + (an − bn ). ∂x1 ∂xn

The Picard iterations Yk (x) defined by Y0 (x) = Y0 = (c1 , . . . , cn ), Yk+1 (x) = Y0 +

Z x x0

F (t, Yk (t))dt,

converge to the unique solution Y and |Y (x) − Yk (x)| ≤ (M/L)ehL hk+1 /(k + 1)!.

47

N-TH ORDER DIFFERENTIAL EQUATIONS

If f1 (x, y1 , . . . , y) , is an L such that

∂fi ∂yj

are continuous in the strip |x − x0 | ≤ a and there

|f (x, Y ) − f (x, Z)| ≤ L|Y − Z| then h can be taken to be a and M = max|f (x, Y0 )|. This happens in the important special case fi (x, y1 , . . . , yn ) = ai1 (x)y1 + · · · + ain (x)yn + bi (x). As a corollary of the above theorem we get the following fundamental theorem for n-th order DE’s.

2.

Theorem of Existence and Uniqueness (II) If f (x, y1 , . . . , yn ) and

∂f ∂yj

are continuous on the box

R : |x − x0 | ≤ a, |yi − ci | ≤ b (1 ≤ i ≤ n) and |f (x, y1 , . . . , yn )| ≤ M on R, then the initial value problem y (n) = f (x, y, y 0 , . . . , y (n−1) ),

y i−1 (x0 ) = ci (1 ≤ 1 ≤ n)

has a unique solution on the interval |x − x0 | ≤ h = max(a, b/M ). Another important application is to the n-th order linear DE a0 (x)y (n) + a1 (x)y (n−1) + · · · + an (x)y = b(x). In this case f1 = y2 , f2 = y3 , fn = p1 (x)y1 + · · · pn (x)yn + q(x) where pi (x) = an−i (x)/a0 (x), q(x) = −b(x)/a0 (x).

3.

Theorem of Existence and Uniqueness (III)

If a0 (x), a1 (x), . . . , an (x) are continuous on an interval I and a0 (x) 6= 0 on I then, for any x0 ∈ I, that is not an endpoint of I, and any scalars c1 , c2 , . . . , cn , the initial value problem a0 (x)y (n) +a1 (x)y (n−1) +· · ·+an (x)y = b(x), has a unique solution on the interval I.

y i−1 (x0 ) = ci (1 ≤ 1 ≤ n)

PART (II): BASIC THEORY OF LINEAR EQUATIONS

In this lecture we give an introduction to several methods for solving higher order differential equations. Most of what we say will apply to the linear case as there are relatively few non-numerical methods for solving nonlinear equations. There are two important cases however where the DE can be reduced to one of lower degree.

3.1

Case (I)

DE has the form: y (n) = f (x, y 0 , y 00 , . . . , y (n−1) ) where on the right-hand side the variable y does not appear. In this case, setting z = y 0 leads to the DE z (n−1) = f (x, z, z 0 , . . . , z (n−2) ) which is of degree n − 1. If this can be solved then one obtains y by integration with respect to x. For example, consider the DE y 00 = (y 0 )2 . Then, setting z = y 0 , we get the DE z 0 = z 2 which is a separable first order equation for z. Solving it we get z = −1/(x + C) or z = 0 from which y = − log(x + C) + D or y = C. The reader will easily verify that there is exactly one of these solutions which satisfies the initial condition y(x0 ) = y0 , y 0 (x0 ) = y00 for any choice of x0 , y0 , y00 which confirms that it is the general solution since the fundamental theorem guarantees a unique solution.

49

50

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

3.2

Case (II)

DE has the form: y (n) = f (y, y 0 , y 00 , . . . , y (n−1) ) where the independent variable x does not appear explicitly on the righthand side of the equation. Here we again set z = y 0 but try for a solution d d z as a function of y. Then, using the fact that dx = z dy , we get the DE µ

z

d dy

¶n−1

µ

¶

(z) = f y, z, z

dz d , . . . , (z )n (z) dy dy

which is of degree n − 1. For example, the DE y 00 = (y 0 )2 is of this type and we get the DE dz z = z2 dy which has the solution z = Cey . Hence y 0 = Cey from which −e−y = Cx + D. This gives y = − log(−Cx − D) as the general solution which is in agreement with what we did previously.

4. 4.1

Linear Equations Basic Concepts and General Properties

Let us now go to linear equations. The general form is L(y) = a0 (x)y (n) + a1 (x)y (n−1) + · · · + an (x)y = b(x). The function L is called a differential operator. The characteristic feature of L is that L(a1 y1 + a2 y2 ) = a1 L(y1 ) + a2 L(y2 ). Such a function L is what we call a linear operator. Moreover, if L1 (y) = a0 (x)y (n) + a1 (x)y (n−1) + · · · + an (x)y L2 (y) = b0 (x)y (n) + b1 (x)y (n−1) + · · · + bn (x)y and p1 (x), p2 (x) are functions of x the function p1 L1 + p2 L2 defined by (p1 L1 + p2 L2 )(y) = p1 (x)L1 (y) + p2 (x)L2 (y) (3.3) = [a0 (x) + p2 (x)b0 (x)] y (n) + · · · [p1 (x)an (x) + p2 (x)bn (x)] y is again a linear differential operator. An important property of linear operators in general is the distributive law: L(L1 + L2 ) = LL1 + LL2 ,

(L1 + L2 )L = L1 L + L2 L.

51

N-TH ORDER DIFFERENTIAL EQUATIONS

The linearity of equation implies that for any two solutions y1 , y2 the difference y1 − y2 is a solution of the associated homogeneous equation L(y) = 0. Moreover, it implies that any linear combination a1 y1 + a2 y2 of solutions y1 , y2 of L(y) = 0 is again a solution of L(y) = 0. The solution space of L(y) = 0 is also called the kernel of L and is denoted by ker(L). It is a subspace of the vector space of real valued functions on some interval I. If yp is a particular solution of L(y) = b(x), the general solution of L(y) = b(x) is ker(L) + yp = {y + yp | L(y) = 0}. The differential operator L(y) = y 0 may be denoted by D. The operator L(y) = y 00 is nothing but D2 = D ◦D where ◦ denotes composition of functions. More generally, the operator L(y) = y (n) is Dn . The identity operator I is defined by I(y) = y. By definition D0 = I. The general linear n-th order ODE can therefore be written h

i

a0 (x)Dn + a1 (x)Dn−1 + · · · + an (x)I (y) = b(x).

5.

Basic Theory of Linear Differential Equations

In this lecture we will develop the theory of linear differential equations. The starting point is the fundamental existence theorem for the general n-th order ODE L(y) = b(x), where L(y) = Dn + a1 (x)Dn−1 + · · · + an (x). We will also assume that a0 (x), a1 (x), . . . , an (x), b(x) are continuous functions on the interval I.

5.1

Basics of Linear Vector Space

5.1.1 Isomorphic Linear Transformation From the fundamental theorem, it is known that for any x0 ∈ I, the initial value problem L(y) = b(x)

y(x0 ) = d1 , y 0 (x0 ) = d2 , . . . , y (n−1) (x0 ) = dn

has a unique solution for any d1 , d2 , . . . , dn ∈ R. Thus, if V is the solution space of the associated homogeneous DE L(y) = 0, the transformation T : V → Rn , defined by T (y) = (y(x0 ), y 0 (x0 ), . . . , y (n−1) (x0 )), is linear transformation of the vector space V into Rn since T (ay + bz) = aT (y) + bT (z).

52

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

Moreover, the fundamental theorem says that T is one-to-one (T (y) = T (z) impliesy = z) and onto (every d ∈ Rn is of the form T (y) for some y ∈ V ). A linear transformation which is one-to-one and onto is called an isomorphism. Isomorphic vector spaces have the same properties.

5.1.2 Dimension and Basis of Vector Space We call the vector space being n-dimensional with the notation by dim(V ) = n. This means that there exists a sequence of elements: y1 , y2 , . . . , yn ∈ V such that every y ∈ V can be uniquely written in the form y = c1 y1 + c2 y2 + . . . cn yn with c1 , c2 , . . . , cn ∈ R. Such a sequence of elements of a vector space V is called a basis for V . In the context of DE’s it is also known as a fundamental set. The number of elements in a basis for V is called the dimension of V and is denoted by dim(V ). If e1 = (1, 0, . . . , 0), e2 = (0, 1, . . . , 0), . . . , en = (0, 0, . . . , 1) is the standard basis of Rn and yi is the unique yi ∈ V with T (yi ) = ei then y1 , y2 , . . . , yn is a basis for V . This follows from the fact that T (c1 y1 + c2 y2 + · · · + cn yn ) = c1 T (y1 ) + c2 T (y2 ) + · · · + cn T (yn ).

5.1.3 (*) Span and Subspace A set of vectors v1 , v2 , · · · , vn in a vector space V is said to span or generate V if every v ∈ V can be written in the form v = c1 v1 + c2 v2 + · · · + cn vn with c1 , c2 , . . . , cn ∈ R. Obviously, not any set of n vectors can span the vector space V . It will be seen that {v1 , v2 , · · · , vn } span the vector space V , if and only if they are linear independent. The set S = span(v1 , v2 , . . . , vn ) = {c1 v1 + c2 v2 + · · · + cn vn | c1 , c2 , . . . , cn ∈ R} consisting of all possible linear combinations of the vectors v1 , v2 , . . . , vn form a subspace of V , which may be also called the span of {v1 , v2 , . . . , vn }. Then V = span(v1 , v2 , . . . , vn ) if and only if v1 , v2 , . . . , vn spans V .

5.1.4 Linear Independency The vectors v1 , v2 , . . . , vn are said to be linearly independent if c1 v1 + c2 v2 + . . . cn vn = 0

53

N-TH ORDER DIFFERENTIAL EQUATIONS

implies that the scalars c1 , c2 , . . . , cn are all zero. A basis can also be characterized as a linearly independent generating set since the uniqueness of representation is equivalent to linear independence. More precisely, c1 v1 + c2 v2 + · · · + cn vn = c01 v1 + c02 v2 + · · · + c0n vn implies

ci = c0i

for all i,

if and only if v1 , v2 , . . . , vn are linearly independent. As an example of a linearly independent set of functions consider cos(x), cos(2x), sin(3x). To prove their linear independence, suppose that c1 , c2 , c3 are scalars such that c1 cos(x) + c2 cos(2x) + c3 sin(3x) = 0 for all x. Then setting x = 0, π/2, π, we get c1 + c2 + c3 = 0, −c2 − c3 = 0, −c1 + c2 =0

(3.4)

from which c1 = c2 = c3 = 0. An example of a linearly dependent set would be sin2 (x), cos2 (x), cos(2x) since cos(2x) = cos2 (x) − sin2 (x) implies that cos(2x) + sin2 (x) + (−1) cos2 (x) = 0.

5.2

Wronskian of n-functions

Another criterion for linear independence of functions involves the Wronskian.

5.2.1 Definition If y1 , y2 , . . . , yn are n functions which have derivatives up to order n − 1 then the Wronskian of these functions is the determinant ¯ ¯ y1 ¯ ¯ y0 ¯ 1 W = W (y1 , y2 , . . . , yn ) = ¯¯ .. ¯. ¯ (n−1) ¯y 1

¯ ¯ ¯ ¯ ¯ ¯. ¯ ¯ (n−1) (n−1) ¯¯ y2 . . . yn

y2 y20 .. .

. . . yn . . . yn0 .. .

If W (x0 ) 6= 0 for some point x0 , then y1 , y2 , . . . , yn are linearly independent. This follows from the fact that W (x0 ) is the determinant of

54

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

the coefficient matrix of the linear homogeneous system of equations in c1 , c2 , . . . , cn obtained from the dependence relation c1 y1 + c2 y2 + · · · + cn yn = 0 and its first n − 1 derivatives by setting x = x0 . For example, if y1 = cos(x), cos(2x), cos(3x) we have ¯ ¯ ¯ ¯ cos(x) cos(2x) cos(3x) ¯ ¯ ¯ W = ¯ − sin(x) −2 sin(2x) −3 sin(3x) ¯¯ ¯ − cos(x) −4 cos(2x) −9 cos(3x) ¯

and W (π/4)) = −8 which implies that y1 , y2 , y3 are linearly independent. Note that W (0) = 0 so that you cannot conclude linear dependence from the vanishing of the Wronskian at a point. This is not the case if y1 , y2 , . . . , yn are solutions of an n-th order linear homogeneous ODE.

5.2.2 Theorem 1 The the Wronskian of n solutions of the n-th order linear ODE L(y) = 0 is subject to the following first order ODE: dW = −a1 (x)W, dx with solution W (x) = W (x0 )e

−

Rx x0

a1 (t)dt

.

From the above it follows that the Wronskian of n solutions of the n-th order linear ODE L(y) = 0 is either identically zero or vanishes nowhere.

5.2.3 Theorem 2 If y1 , y2 , . . . , yn are solutions of the linear ODE L(y) = 0, the following are equivalent: 1 y1 , y2 , . . . , yn is a basis for the vector space V = ker(L); 2 y1 , y2 , . . . , yn are linearly independent; 3

(∗)

y1 , y2 , . . . , yn span V ;

4 y1 , y2 , . . . , yn generate ker(L); 5 W (y1 , y2 , . . . , yn ) 6= 0 at some point x0 ; 6 W (y1 , y2 , . . . , yn ) is never zero.

55

N-TH ORDER DIFFERENTIAL EQUATIONS

Proof. The equivalence of 1, 2, 3 follows from the fact that ker(L) is isomorphic to Rn . The rest of the proof follows from the fact that if the Wronskian were zero at some point x0 the homogeneous system of equations c1 y1 (x0 ) + c1 y2 (x0 ) + · · · + cn yn (x0 ) c1 y10 (x0 ) + c1 y20 (x0 ) + · · · + cn yn0 (x0 ) .. . (n−1)

c1 y1

(n−1)

(x0 ) + c1 y2

=0 =0 (n−1)

(x0 ) + · · · + cn yn

(3.5)

(x0 ) = 0

would have a non-zero solution for c1 , c2 , . . . , cn which would imply that c1 y1 + c2 y2 + · · · + cn yn = 0 and hence that y1 , y2 , . . . , yn are not linearly independent.

QED

From the above, we see that to solve the n-th order linear DE L(y) = b(x) we first find linear n independent solutions y1 , y2 , . . . , yn of L(y) = 0. Then, if yP is a particular solution of L(y) = b(x), the general solution of L(y) = b(x) is y = c1 y1 + c2 y2 + · · · + cn yn + yP . (n−1)

The initial conditions y(x0 ) = d1 , y 0 (x0 ) = d2 , . . . , yn determine the constants c1 , c2 , . . . , cn uniquely.

(x0 ) = dn then

PART (III): SOLUTIONS FOR EQUATIONS WITH CONSTANTS COEFFICIENTS (1)

In what follows, we shall first focus on the linear equations with constant coefficients: L(y) = a0 y (n) + a1 y (n−1) + · · · + an y = b(x) and present two different approaches to solve them.

6.

The Method with Undetermined Parameters

To illustrate the idea, as a special case, let us first consider the 2-nd order Linear equation with the constant coefficients: L(y) = ay 00 + by 0 + cy = f (x).

(3.6)

The associate homogeneous equation is: L(y) = ay 00 + by 0 + cy = 0.

6.1

(3.7)

Basic Equalities (I)

We first give the following basic identities: D(erx ) = rerx ; D2 (erx ) = r2 erx ; · · · Dn (erx ) = rn erx .

(3.8)

To solve this equation, we assume that the solution is in the form y(x) = erx , where r is a constant to be determined. Due to the properties of the exponential function erx : y 0 (x) = ry(x); y 00 (x) = r2 y(x); · · · y (n) = rn y(x),

57

58

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

we can write L(erx ) = φ(r)erx .

(3.9)

for any given (r, x), where φ(r) = ar2 + br + c. is called the characteristic polynomial. From (3.9) it is seen that the function erx satisfies the equation (3.6), namely L(erx ) = 0, as long as the constant r is the root of the characteristic polynomial, i.e. φ(r) = 0. In general, the polynomial φ(r) has two roots (r1, r2 ): One can write φ(r) = ar2 + br + c = a(r − r1 )(r − r2 ). Accordingly, the equation (3.7) has two solutions: {y1 (x) = er1 x ; y2 (x) = er2 x }. Two cases should be discussed separately.

6.2

Cases (I) ( r1 > r2 )

When b2 − 4ac > 0, the polynomial φ(r) has two distinct real roots (r1 6= r2 ). In this case, the two solutions, y( x); y2 (x) are different. The following linear combination is not only solution, but also the general solution of the equation: y(x) = Ay1 (x) + By2 (x),

(3.10)

where A, B are arbitrary constants. To prove that, we make use of the fundamental theorem which states that if y, z are two solutions such that y(0) = z(0) = y0 and y 0 (0) = z 0 (0 = y00 ) then y = z. Let y be any solution and consider the linear equations in A, B Ay1 (0) + By2 (0) = y(0), Ay10 (0) + By20 (0) = y 0 (0),

(3.11)

A+B = y0 , Ar1 + Br2 = y00 .

(3.12)

or

Due to r1 6= r2 , these conditions leads to the unique solution A, B. With this choice of A, B the solution z = Ay1 + By2 satisfies z(0) = y(0), z 0 (0) = y 0 (0) and hence y = z. Thus, (3.10) contains all possible solutions of the equation, so, it is indeed the general solution.

59

N-TH ORDER DIFFERENTIAL EQUATIONS

6.3

Cases (II) ( r1 = r2 )

When b2 −4ac = 0, the polynomial φ(r) has double root: r1 = r2 = −b 2a . In this case, the solution y1 (x) = y2 (x) = er1 x . Thus, for the general solution, one needs to derive another type of the second solution. For this purpose, one may use the method of reduction of order. Let us look for a solution of the form C(x)er1 x with the undetermined function C(x). By substituting the equation, we derive that ³

´

h

i

L C(x)er1 x = C(x)φ(r1 )er1 x +a C 00 (x)+2r1 C 0 (x) er1 x +bC 0 (x)er1 x = 0. Noting that φ(r1 ) = 0; we get

2ar1 + b = 0,

C 00 (x) = 0

or C(x) = Ax + B, where A, B are arbitrary constants. Thus, we solution: y(x) = (Ax + B)er1 x ,

(3.13)

is a two parameter family of solutions consisting of the linear combinations of the two solutions y1 = er1 x and y2 = xer1 x . It is also the general solution of the equation. The proof is similar to that given for the case (I) based on the fundamental theorem of existence and uniqueness. Let y be any solution and consider the linear equations in A, B Ay1 (0) + By2 (0) = y(0), Ay10 (0) + By20 (0) = y 0 (0),

(3.14)

A = y(0), Ar1 + B = y 0 (0).

(3.15)

or

these conditions leads to the unique solution A = y(0), B = y 0 (0) − r1 y(0). With this choice of A, B the solution z = Ay1 + By2 satisfies z(0) = y(0), z 0 (0) = y 0 (0) and hence y = z. Thus, (3.13) contains all possible solutions of the equation, so, it is indeed the general solution. The approach presented in this subsection is applicable to any higher order equations with constant coefficients. Example 1. Consider the linear DE y 00 + 2y 0 + y = x. Here L(y) = y 00 + 2y 0 + y. A particular solution of the DE L(y) = x is yp = x − 2. The associated homogeneous equation is y 00 + 2y 00 + y = 0.

60

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

The characteristic polynomial φ(r) = r2 + 2r + 1 = (r + 1)2 has double roots r1 = r2 = −1. Thus the general solution of the DE y 00 + 2y 0 + y = x is y = Axe−x + Be−x + x − 2. This equation can be solved quite simply without the use of the fundamental theorem if we make essential use of operators.

6.4

Cases (III) ( r1;2 = λ ± iµ)

When b2 − 4ac < 0, the polynomial φ(r) has two conjugate complex roots r1,2 = λ ± iµ. We have to define the complex number, i2 = −1;

i3 = −i;

i4 = 1;

i5 = i, · · ·

and define and complex function with the Taylor series: eix =

∞ n n X i x n=0

n!

=

∞ X (−1)n x2n n=0

2n!

+i

∞ X (−1)n x2n+1 n=0

(2n + 1)!

= cos x + i sin x.

(3.16)

From the definition, it follows that ex+iy = ex eiy = ex (cos y + i sin y) . and

D(erx ) = rex ,

Dn (erx ) = rn ex

where r is a complex number. So that, the basic equalities are now extended to the case with complex number r. Thus, we have the two complex solutions: y1 (x) = er1 x = eλx (cos µx+i sin µx),

y2 (x) = er2 x = eλx (cos µx−i sin µx)

with a proper combination of these two solutions, one may derive two real solutions: y˜1 (x) = eλx cos µx,

y˜2 (x) = eλx sin µx

and the general solution: y(x) = eλx (A cos µx + B sin µx).

PART (IV): SOLUTIONS FOR EQUATIONS WITH CONSTANTS COEFFICIENTS (2)

We adopt the differential operator D and write the linear equation in the following form: L(y) = (a0 D(n) + a1 D(n−1) + · · · + an )y = P (D)y = b(x).

7. 7.1

The Method with Differential Operator Basic Equalities (II).

We may prove the following basic identity of differential operators: for any scalar a, (D − a) = eax De−ax (D − a)n = eax Dn e−ax

(3.17)

where the factors eax , e−ax are interpreted as linear operators. This identity is just the fact that µ ¶ dy d −ax − ay = eax (e y) . dx dx The formula (3.17) may be extensively used in solving the type of linear equations under discussion. Let write the equation (3.7) with the differential operator in the following form: L(y) = (aD2 + bD + c)y = φ(D)y = 0, where

(3.18)

φ(D) = (aD2 + bD + c) is a polynomial of D. We now re-consider the cases above-discussed with the previous method.

61

62

7.2

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

Cases (I) ( b2 − 4ac > 0)

The polynomial φ(r) have two distinct real roots r1 > r2 . Then, we can factorize the polynomial φ(D) = (D − r1 )(D − r2 ) and re-write the equation as: L(y) = (D − r1 )(D − r2 )y = 0. letting z = (D − r2 )y, in terms the basic equalities, we derive (D − r1 )z = er1 x De−r1 x z = 0, er1 x z = A,

z = Aer1 x .

Furthermore, from (D − r2 )y = er2 x De−r2 x y = z = Aer1 x , we derive and

D(e−r2 x y) = ze−r2 x = Ae(r1 −r2 )x ˜ r1 x + Ber2 x , y = Ae

A where A˜ = (r1 −r , B are arbitrary constants. It is seen that, in general, 2) to solve the equation

L(y) = (D − r1 )(D − r2 ) · · · (D − rn )y = 0, where ri 6= rj , (i 6= j), one can first solve each factor equations (D − ri )yi = 0,

(i = 1, 2, · · · , n)

separately. The general solution can be written in the form: y(x) = y1 (x) + y2 (x) + · · · + yn (x).

7.3

Cases (II) ( b2 − 4ac = 0)

. The polynomial φ(r) have double real roots r1 = r2 . Then, we can factorize the polynomial φ(D) = (D − r1 )2 and re-write the equation as: L(y) = (D − r1 )2 y = 0. In terms the basic equalities, we derive (D − r1 )2 y = er1 x D2 e−r1 x y = 0,

N-TH ORDER DIFFERENTIAL EQUATIONS

hence,

63

D2 (e−r1 x y) = 0.

One can solve

(e−r1 x y) = A + Bx,

or

y = (A + Bx)er1 x . In general, for the equation, L(y) = (D − r1 )n y = 0.

we have the general solution: y = (A1 + A2 x + · · · + An xn−1 )er1 x . So, we may write ker((D − a)n ) = {(a0 + ax + · · · + an−1 xn−1 )eax | a0 , a1 , . . . , an−1 ∈ R}.

7.4

Cases (III) ( b2 − 4ac < 0)

The polynomial φ(r) have two complex conjugate roots r1,2 = λ ± iµ. Then, we can factorize the polynomial φ(D) = (D−λ)2 +µ2 , and re-write the equation as: L(y) = ((D − λ)2 + µ2 )y = 0. Let us consider the special case first: L(z) = (D2 + µ2 )z = 0. From the formulas: D(cos µx) = −µ sin x,

D(sin x) = µ cos x,

it follows that z(x) = A cos µx + B sin µx. To solve for y(x), we re-write the equation (3.19) as (eλx D2 e−λx + µ2 )y = 0. Then

D2 (e−λx y) + µ2 e−λx y = (D2 + µ2 )e−λx y = 0.

Thus, we derive e−λx y(x) = A cos µx + B sin µx,

(3.19)

64

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

or y(x) = eλx (A cos µx + B sin µx).

(3.20)

One may also consider case (I) with the complex number r1 , r2 and obtain the complex solution: y(x) = eλx (Aeiµx + Be−iµx ).

7.5

(3.21)

Theorems

In summary, it can be proved that the following results hold: ker ((D − a)m ) = span(eax , xeax , . . . , xm−1 eax ) It means that ((D − a)m )y = 0 has a set of fundamental solutions: n

eax , xeax , . . . , xm−1 eax

o

ker ((D − a)2 + b2 )m ) = span(eax f (x), xeax f (x), . . . , xm−1 eax f (x)), f (x) = cos(bx) or sin(bx) It means that ((D − a)2 + b2 )m )y = 0 has a set of fundamental solutions: n o eax f (x), xeax f (x), . . . , xm−1 eax f (x) , where f (x) = cos(bx) or sin(bx). ker(P (D)Q(D)) = ker(P (D))+ker(Q(D)) = {y1 +y2 | y1 ∈ ker(P (D)), y2 ∈ ker(Q(D))}, if P (X), Q(X) are two polynomials with constant coefficients that have no common root. It means that if P (X), Q(X) have no common roots, then the set of fundamental solutions for the operator P (D)Q(D) is just the joint set: {p1 (x), p2 (x), · · · , pn (x); q1 (x), q2 (x), · · · qm (x)}, where {p1 (x), p2 (x), · · · , pn (x)} is the set of fundamental solutions for the operator P (D), and {q1 (x), q2 (x), · · · qm (x)} is the set of fundamental solutions for the operator Q(D). Example 1. By using the differential operation method, one can easily solve some inhomogeneous equations. For instance, let us reconsider the example 1. One may write the DE y 00 + 2y 0 + y = x in the operator form as (D2 + 2D + I)(y) = x.

N-TH ORDER DIFFERENTIAL EQUATIONS

65

The operator (D2 + 2D + I) = φ(D) can be factored as (D + I)2 . With (3.17), we derive that (D + I)2 = (e−x Dex )(e−x Dex ) = e−x D2 ex . Consequently, the DE (D + I)2 (y) = x can be written e−x D2 ex (y) = x or d2 x (e y) = xex dx which on integrating twice gives ex y = xex − 2ex + Ax + B,

y = x − 2 + Axe−x + Be−x .

We leave it to the reader to prove that ker((D − a)n ) = {(a0 + ax + · · · + an−1 xn−1 )eax | a0 , a1 , . . . , an−1 ∈ R}. Example 2. Now consider the DE y 00 − 3y 0 + 2y = ex . In operator form this equation is (D2 − 3D + 2I)(y) = ex . Since (D2 − 3D + 2I) = (D − I)(D − 2I), this DE can be written (D − I)(D − 2I)(y) = ex . Now let z = (D − 2I)(y). Then (D − I)(z) = ex , a first order linear DE which has the solution z = xex + Aex . Now z = (D − 2I)(y) is the linear first order DE y 0 − 2y = xex + Aex which has the solution y = ex − xex − Aex + Be2x . Notice that −Aex + Be2x is the general solution of the associated homogeneous DE y 00 − 3y 0 + 2y = 0 and that ex − xex is a particular solution of the original DE y 00 − 3y 0 + 2y = ex . Example 3. Consider the DE y 00 + 2y 0 + 5y = sin(x) which in operator form is (D2 + 2D + 5I)(y) = sin(x). Now D2 + 2D + 5I = (D + I)2 + 4I and so the associated homogeneous DE has the general solution Ae−x cos(2x) + Be−x sin(2x).

66

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

All that remains is to find a particular solution of the original DE. We leave it to the reader to show that there is a particular solution of the form C cos(x) + D sin(x). Example 4. Solve the initial value problem y 000 − 3y 00 + 7y 0 − 5y = 0,

y(0) = 1, y 0 (0) = y 00 (0) = 0.

The DE in operator form is (D3 − 3D2 + 7D − 5)(y) = 0. Since φ(r) = r3 − 3r2 + 7r − 5 = (r − 1)(r2 − 2r + 5) = (r − 1)[(r − 1)2 + 4], we have L(y) = = = =

(D3 − 3D2 + 7D − 5)(y) (D − 1)[(D − 1)2 + 4](y) [(D − 1)2 + 4](D − 1)(y) 0.

(3.22)

From here, it is seen that the solutions for (D − 1)(y) = 0,

(3.23)

y(x) = c1 ex ,

(3.24)

[(D − 1)2 + 4](y) = 0,

(3.25)

y(x) = c2 ex cos(2x) + c3 ex sin(2x),

(3.26)

namely, and the solutions for namely, must be the solutions for our equation (3.22). Thus, we derive that the following linear combination y = c1 ex + c2 ex cos(2x) + c3 ex sin(2x),

(3.27)

must be the solutions for our equation (3.22). In solution (3.27), there are three arbitrary constants (c1 , c2 , c3 ). One can prove that this solution is the general solution, which covers all possible solutions of (3.22). For instance, given the I.C.’s: y(0) = 1, y 0 (0) = 0, y 00 (0) = 0, from (3.27), we can derive c1 + c2 = 1, c1 + c2 + 2c3 = 0, c1 − 3c2 + 4c3 = 0, and find c1 = 5/4, c2 = −1/4, c3 = −1/2.

PART (V): FINDING A PARTICULAR SOLUTION FOR INHOMOGENEOUS EQUATION

Variation of parameters is method for producing a particular solution of a special kind for the general linear DE in normal form L(y) = y (n) + a1 (x)y (n−1) + · · · + an (x)y = b(x) from a fundamental set y1 , y2 , . . . , yn of solutions of the associated homogeneous equation.

8.

The Differential Operator for Equations with Constant Coefficients Given L(y) = P (D)(y) = (a0 D(n) + a1 D(n−1) + · · · + an D)y = b(x).

Assume that the inhomogeneous term b(x) is a solution of linear equation: Q(D)(b(x)) = 0. Then we can transform the original inhomogeneous equation to the homogeneous equation by applying the differential operator Q(D) to its both sides, Φ(D)(y) = Q(D)P (D)(y) = 0. The operator Q(D) is called the Annihilator. The above method is also called the Annihilator Method. Example 1. Solve the initial value problem y 000 − 3y 00 + 7y 0 − 5y = x + ex ,

67

y(0) = 1, y 0 (0) = y 00 (0) = 0.

68

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

This DE is non-homogeneous. The associated homogeneous equation was solved in the previous lecture. Note that in this example, In the inhomogeneous term b(x) = x + ex is in the kernel of Q(D) = D2 (D − 1). Hence, we have D2 (D − 1)2 ((D − 1)2 + 4)(y) = 0 which yields y = Ax+B +Cxex +c1 ex +c2 ex cos(2x)+c3 ex sin(2x). This shows that there is a particular solution of the form yP = Ax+B +Cxex which is obtained by discarding the terms in the solution space of the associated homogeneous DE. Substituting this in the original DE we get y 000 − 3y 00 + 7y 0 − 5y = 7A − 5B − 5Ax − Cex which is equal to x + ex if and only if 7A − 5B = 0, −5A = 1, −C = 1 so that A = −1/5, B = −7/25, C = −1. Hence the general solution is y = c1 ex + c2 ex cos(2x) + c3 ex sin(2x) − x/5 − 7/25 − xex . To satisfy the initial condition y(0) = 0, y 0 (0) = y 00 (0) = 0 we must have c1 + c2 = 32/25, c1 + c2 + 2c3 = 6/5, c1 − 3c2 + 4c3 = 2

(3.28)

which has the solution c1 = 3/2, c2 = −11/50, c3 = −1/25. It is evident that if the function b(x) can not be eliminated by any linear operator Q(D), the annihilator method will not applicable.

9.

The Method of Variation of Parameters In this method we try for a solution of the form yP = C1 (x)y1 + C2 (x)y2 + · · · + Cn (x)yn .

Then yP0 = C1 (x)y10 + C2 (x)y20 + · · · + Cn (x)yn0 + C10 (x)y1 + C20 (x)y2 + · · · + Cn0 (x)yn and we impose the condition C10 (x)y1 + C20 (x)y2 + · · · + Cn0 (x)yn = 0. Then yP0 = C1 (x)y10 + C2 (x)y20 + · · · + Cn (x)yn0 and hence yP00 = C1 (x)y100 +C2 (x)y200 +· · ·+Cn (x)yn00 +C10 (x)y10 +C20 (x)y20 +· · ·+Cn0 (x)yn0 . Again we impose the condition C10 (x)y10 + C20 (x)y20 + · · · + Cn0 (x)yn0 = 0 so that yP00 = C1 (x)y100 + C2 (x)y200 + · · · + Cn (x)yn0 .

69

N-TH ORDER DIFFERENTIAL EQUATIONS

We do this for the first n − 1 derivatives of y so that for 1 ≤ k ≤ n − 1 (k)

(k)

(k)

yP = C1 (x)y1 + C2 (x)y2 + · · · Cn (x)yn(k) , (k)

(k)

C10 (x)y1 + C20 (x)y2 + · · · + Cn0 (x)yn(k) = 0. (n−1)

Now substituting yP , yP0 , . . . , yP

in L(y) = b(x) we get (n−1)

C1 (x)L(y1 ) + C2 (x)L(y2 ) + · · · + Cn (x)L(yn ) + C10 (x)y1 (n−1) +C20 (x)y2

(3.29)

+ · · · + Cn0 (x)yn(n−1) = b(x).

But L(yi ) = 0 for 1 ≤ k ≤ n so that (n−1)

C10 (x)y1

(n−1)

+ C20 (x)y2

+ · · · + Cn0 (x)yn(n−1) = b(x).

We thus obtain the system of n linear equations for C10 (x), . . . , Cn0 (x) C10 (x)y1 + C20 (x)y2 + · · · + Cn0 (x)yn C10 (x)y10 + C20 (x)y20 + · · · + Cn0 (x)yn0 .. . (n−1)

C10 (x)y1

(n−1)

+ C20 (x)y2

= 0, = 0, (n−1)

+ · · · + Cn0 (x)yn

(3.30)

= b(x).

If we solve this system using Cramer’s Rule and integrate, we find Ci (x) =

Z x x0

(−1)n+i b(t)

Wi dt W

where W = W (y1 , y2 , . . . , yn ) and Wi = W (y1 . . . , yˆi , . . . , yn ) where theˆ means that yi is omitted. Note that the particular solution yP found in this way satisfies (n−1)

yP (x0 ) = yP0 (x0 ) = · · · = yP

= 0.

The point x0 is any point in the interval of continuity of the ai (x) and b(x). Note that yP is a linear function of the function b(x). Example 2. Find the general solution of y 00 + y = 1/x on x > 0. The general solution of y 00 + y = 0 is y = c1 cos(x) + c2 sin(x). Using variation of parameters with y1 = cos(x), y2 = sin(x), b(x) = 1/x and x0 = 1, we have W = 1, W1 = sin(x), W2 = cos(x) and we obtain the particular solution yp = C1 (x) cos(x) + C2 (x) sin(x) where C1 (x) = −

Z x sin(t) 1

t

dt,

C2 (x) =

Z x cos(t) 1

t

dt.

70

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

The general solution of y 00 + y = 1/x on x > 0 is therefore y = c1 cos(x) + c2 sin(x) −

µZ x sin(t)

t

1

¶

dt cos(x) +

µZ x cos(t)

t

1

¶

dt sin(x).

When applicable, the annihilator method is easier as one can see from the DE y 00 + y = ex . Here it is immediate that yp = ex /2 is a particular solution while variation of parameters gives yp = −

µZ x 0

¶ t

e sin(t)dt cos(x) +

µZ x 0

¶ t

e cos(t)dt sin(x).

The integrals can be evaluated using integration by parts: Rx t Rx t x 0 e cos(t)dt = e cos(x) − 1 + 0 e sin(t)dt R

= ex cos(x) + ex sin(x) − 1 −

x t 0 e cos(t)dt

(3.31)

which gives Z x 0

Z x 0

h

i

et cos(t)dt = ex cos(x) + ex sin(x) − 1 /2

et sin(t)dt = ex sin(x) −

Z x 0

h

i

et cos(t)dt = ex sin(x) − ex cos(x) + 1 /2

so that after simplification yp = ex /2 − cos(x)/2 − sin(x)/2.

PART (VI): LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS

In this lecture we will give a few techniques for solving certain linear differential equations with non-constant coefficients. We will mainly restrict our attention to second order equations. However, the techniques can be extended to higher order equations. The general second order linear DE is p0 (x)y 00 + p1 (x)y 0 + p2 (x)y = q(x). This equation is called a non-constant coefficient equation if at least one of the functions pi is not a constant function.

10.

Euler Equations

An important example of a non-constant linear DE is Euler’s equation x2 y 00 + axy 0 + by = 0, where a, b are constants. This equation has singularity at x = 0. The fundamental theorem of existence and uniqueness of solution holds in the region x > 0 and x, 0, respectively. So one must solve the problem in the region x > 0, or x < 0 separately. We first consider the region x > 0. This Euler equation can be transformed into a constant coefficient DE by the change of independent variable x = et . This is most easily seen by noting that dy dx dy dy = = et = xy 0 dt dx dt dx so that

dy dx

= e−t dy dt . In operator form, we have d d d = et =x . dt dx dx

71

72

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

If we set D =

d dt ,

we have

d dx

= e−t D so that

d2 = e−t De−t D = e−2t et De−t D = e−2t (D − 1)D dx2 so that x2 y 00 = D(D − 1). By induction one easily proves that dn = e−nt D(D − 1) · · · (D − n + 1) dxn or xn y (n) = D(D − 1) · · · (D − n + 1)(y). With the variable t, Euler’s equation becomes d2 y dy + (a − 1) + by = q(et ), 2 dt dt which is a linear constant coefficient DE. Solving this for y as a function of t and then making the change of variable t = ln(x), we obtain the solution of Euler’s equation for y as a function of x. For the region x < 0, we may let −x = et , or |x| = et . Then the equation x2 y 00 + axy 0 + by = 0, (x < 0) is changed to the same form d2 y dy + (a − 1) + by = 0. dt2 dt Hence, we have the solution y(t) = y(ln |x|) (x < 0). The above approach, can extend to solve the n-th order Euler equation xn y (n) + a1 xn−1 y (n−1) + · · · + an y = q(x), where a1 , a2 , . . . an are constants. Example 1. Solve x2 y 00 + xy 0 + y = ln(x), (x > 0). Making the change of variable x = et we obtain d2 y +y =t dt2 whose general solution is y = A cos(t) + B sin(t) + t. Hence y = A cos(ln(x)) + B sin(ln(x)) + ln(x) is the general solution of the given DE.

73

N-TH ORDER DIFFERENTIAL EQUATIONS

Example 2. Solve x3 y 000 + 2x2 y 00 + xy 0 − y = 0,

(x > 0).

This is a third order Euler equation. Making the change of variable x = et , we get ³

´

D(D − 1)(D − 2) + 2D(D − 1) + (D − 1) (y) = (D − 1)(D2 + 1)(y) = 0

which has the general solution y = c1 et + c2 sin(t) + c3 cos(t). Hence y = c1 x + c2 sin(ln(x)) + c3 cos(ln(x)) is the general solution of the given DE.

11.

Exact Equations

The DE p0 (x)y 00 + p1 (x)y 0 + p2 (x)y = q(x) is said to be exact if p0 (x)y 00 + p1 (x)y 0 + p2 (x)y =

d (A(x)y 0 + B(x)y). dx

In this case the given DE is reduced to solving the linear DE Z

A(x)y 0 + B(x)y =

q(x)dx + C

a linear first order DE. The exactness condition can be expressed in operator form as p0 D2 + p1 D + p2 = D(AD + B). d Since dx (A(x)y 0 + B(x)y) = A(x)y 00 + (A0 (x) + B(x))y 0 + B 0 (x)y, the exactness condition holds if and only if A(x), B(x) satisfy

A(x) = p0 (x),

B(x) = p1 (x) − p00 (x),

B 0 (x) = p2 (x).

Since the last condition holds if and only if p01 (x) − p000 (x) = p2 (x), we see that the given DE is exact if and only if p000 − p01 + p2 = 0 in which case p0 (x)y 00 + p1 (x)y 0 + p2 (x)y =

d (p0 (x)y 0 + (p1 (x) − p00 (x))y). dx

Example 3. Solve the DE xy 00 + xy 0 + y = x,

(x > 0).

74

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

This is an exact equation since the given DE can be written d (xy 0 + (x − 1)y) = x. dx Integrating both sides, we get xy 0 + (x − 1)y = x2 /2 + A which is a linear DE. The solution of this DE is left as an exercise.

12.

Reduction of Order

If y1 is a non-zero solution of a homogeneous linear n-th order DE, one can always find a second solution of the form y = C(x)y1 where C 0 (x) satisfies a homogeneous linear DE of order n − 1. Since we can choose C 0 (x) 6= 0 we find in this way a second solution y2 = C(x)y1 which is not a scalar multiple of y1 . In particular for n = 2, we obtain a fundamental set of solutions y1 , y2 . Let us prove this for the second order DE p0 (x)y 00 + p1 (x)y 0 + p2 (x)y = 0. If y1 is a non-zero solution we try for a solution of the form y = C(x)y1 . Substituting y = C(x)y1 in the above we get ³

´

³

´

p0 (x) C 00 (x)y1 +2C 0 (x)y10 +C(x)y100 +p1 (x) C 0 (x)y1 +C(x)y10 +p2 (x)C(x)y1 = 0. Simplifying, we get p0 y1 C 00 (x) + (p0 y10 + p1 y1 )C 0 (x) = 0 since p0 y100 + p1 y10 + p2 y1 = 0. This is a linear first order homogeneous DE for C 0 (x). Note that to solve it we must work on an interval where y1 (x) 6= 0. However, the solution found can always be extended to the places where y1 (x) = 0 in a unique way by the fundamental theorem. The above procedure can also be used to find a particular solution of the non-homogenous DE p0 (x)y 00 + p1 (x)y 0 + p2 (x)y = q(x) from a non-zero solution of p0 (x)y 00 + p1 (x)y 0 + p2 (x)y = 0. Example 4. Solve y 00 + xy 0 − y = 0. Here y = x is a solution so we try for a solution of the form y = C(x)x. Substituting in the given DE, we get C 00 (x)x + 2C 0 (x) + x(C 0 (x)x + C(x)) − C(x)x = 0 which simplifies to xC 00 (x) + (x2 + 2)C 0 (x) = 0.

75

N-TH ORDER DIFFERENTIAL EQUATIONS

Solving this linear DE for C 0 (x), we get C 0 (x) = Ae−x so that

Z

2 /2

/x2

dx

C(x) = A

x2 ex2 /2

+B

Hence the general solution of the given DE is Z

y = A1 x + A2 x

dx x2 ex2 /2

.

Example 5. Solve y 00 + xy 0 − y = x3 ex . By the previous example, the general solution of the associated homogeneous equation is Z

yH = A1 x + A2 x

dx x2 ex2 /2

.

Substituting yp = xC(x) in the given DE we get C 00 (x) + (x + 2/2)C 0 (x) = x2 ex . Solving for C 0 (x) we obtain µ

1 C (x) = 2 x2 /2 A2 + x e 0

where H(x) =

Z

4 x+x2 /2

x e

1 x2 ex2 /2

This gives

Z

C(x) = A1 + A2

¶

dx = A2

Z

x4 ex+x

dx + 2 x ex2 /2

2 /2

1 x2 ex2 /2

+ H(x),

dx.

Z

H(x)dx,

We can therefore take Z

yp = x

H(x)dx,

so that the general solution of the given DE is Z

y = A1 x + A2 x

dx x2 ex2 /2

+ yp (x) = yH (x) + yp (x).

PART (VII): SOME APPLICATIONS OF SECOND ORDER DE’S

13.

(*) Vibration System

We now give an application of the theory of second order DE’s to the description of the motion of a simple mass-spring mechanical system with a damping device. We assume that the damping force is proportional to the velocity of the mass. If there are no external forces we obtain the differential equation d2 x dx + kx = 0 +b 2 dt dt where x = x(t) is the displacement from equilibrium at time t of the mass of m > 0 units, b ≥ 0 is the damping constant and k > 0 is the spring d constant. In operator form with D = dt this DE is, after normalizing, m

µ

¶

b k D + D+ (x) = 0. m m 2

The characteristic polynomial r2 + (b/m)r + k/m has discriminant ∆ = (b2 − 4km)/m2 . If b2 < 4km we have ∆ < 0 and the characteristic polynomial factorizes in the form (r + b/2m)2 + ω 2 with ω=

s

p

4km − b2 /2m =

k − (b/2m)2 . m

In this case the characteristic polynomial has complex roots −b/2m ± iω and the general solution of the DE is y = e−bt/2m (A cos(ωt) + B sin(ωt) = Ce−bt/2m sin(ωt + θ)

77

78

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

√ where C = A2 + B 2 and 0 ≤ θ ≤ 2π defined by cos(θ) = A/C, sin(θ) = B/C. The angle θ is called the phase. In this case we see that the mass oscillates with frequency ω/2π and decreasing amplitude. If b = 0 there is no damping and the mass oscillates with frequency ω/2π and constant amplitude; such motion is called simple harmonic. If b2 ≥ 4km we have ∆ ≥ 0 and so the characteristic polynomial has real roots r1 = −b/2m +

p

b2 − 4km/2m,

r2 = −b/2m −

p

b2 − 4km/2m

which are both negative. If r1 = r2 = r the general solution of our DE is y = Aert + Btert and if r1 6= r2 the general solution is y = Aer1 t + Ber2 t . In both cases y → 0 as t → ∞. In the second case we have what is called over damping and in the first case the over damping is said to be critical. In each the mass returns to its equilibrium position without oscillations. Suppose now that our mass-spring system is subject to an external force so that our DE now becomes m

d2 x dx + kx = F (t). +b 2 dt dt

The function F (t) is called the forcing function and measures the magnitude and direction of the external force. We consider the important special case where the forcing function is harmonic F (f ) = F0 cos(γt),

F0 > 0 a constant.

We also assume that we have under-damping with damping constant b > 0. In this case the DE has a particular solution of the form yp = A1 cos(γt) + A2 sin(γt). Substituting the the DE and simplifying, we get ((k−mγ 2 )A1 +bγA2 ) cos(γt)+(−bγA1 +(k−mγ 2 )A2 ) sin(γt) = F0 cos(γt). Setting the corresponding coefficients on both sides equal, we get (k − mγ 2 )A1 + bγA2 = F0 , −bγA1 + (k − mγ 2 )A2 = 0.

(3.32)

79

N-TH ORDER DIFFERENTIAL EQUATIONS

Solving for A1 , A2 we get A1 =

F0 (k − mγ 2 ) , (k − mγ 2 )2 + b2 γ 2

A2 =

F0 bγ (k − mγ 2 )2 + b2 γ 2

and yp = (k−mγF2 )02 +b2 γ 2 ((k − mγ 2 ) cos(γt) + bγ sin(γt)) F0 sin(γt + φ). =√ 2 2 2 2

(3.33)

(k−mγ ) +b γ

The general solution of our DE is then y = Ce−bt/2m sin(ωt + θ) + p

F0 sin(γt + φ). (k − mγ 2 )2 + b2 γ 2

Because of damping the first term tends to zero and is called the transient part of the solution. The second term, the steady-state part of the solution, is due to the presence of the forcing function F0 cos(γt). It is harmonic with the same frequency γ/2π but is out of phase with it by an angle φ − π/2. The ratio of the magnitudes 1 (k − mγ 2 )2 + b2 γ 2

M (γ) = p

is called the gain factor. The graph of the function M (γ) is called the resonance curve. It has a maximum of 1 q

b q

k m

−

b2 4m2

2

k b when γ = γr = m − 2m 2 . The frequency γr /2π is called the resonance frequency of the system. When γ = γr the system is said to be in resonance with the external force. Note that M (γr ) gets arbitrarily large as b → 0. We thus see that the system is subject to large oscillations if the damping constant is very small and the forcing function has a frequency near the resonance frequency of the system.

The above applies to a simple LRC electrical circuit where the differential equation for the current I is d2 I dI + R + I/C = F (t) 2 dt dt where L is the inductance, R is the resistance and C is the capacitance. The resonance phenomenon is the reason why we can send and receive and amplify radio transmissions sent simultaneously but with different frequencies. L

Chapter 4 SERIES SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS

81

PART (I): SERIES SOLUTIONS NEAR A ORDINARY POINT

A function f (x) of one variable x is said to be analytic at a point x = x0 if it has a convergent power series expansion f (x) =

∞ X

an (x−x0 )n = a0 +a1 (x−x0 )+a2 (x−x0 )2 +· · ·+an (x−x0 )n +· · ·

0

for |x − x0 | < R, R > 0. This point x = x0 is also called ordinary point. Otherwise, f (x) is said to have a singularity at x = x0 . The largest such R (possibly +∞) is called the radius of convergence of the power series. The series converges for every x with |x − x0 | < R and diverges for every x with |x − x0 | > R. There is a formula for R = 1` , where 1 |an+1 | `= , or lim , 1/n n→∞ |an | lim |an | n→∞

if the latter limit exists. The same is true if x, x0 , ai are complex. For example, 1 = 1 − x2 + x4 − x6 + · · · + (−1)n x2n + · · · 1 + x2 for |x| < 1. The radius of convergence of the series is 1. It is also equal to the distance from 0 to the nearest singularity x = i of 1/(x2 + 1) in the complex plane. Power series can be integrated and differentiated within the interval (disk) of convergence. More precisely, for |x − x0 | < R we have f 0 (x) =

∞ X

nan xn−1 =

n=1

∞ X

(n + 1)an+1 xn ,

n=0

83

84

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS Z xX ∞ 0 n=0

an tn =

∞ X

∞ an n+1 X an−1 n x = x n+1 n n=0 n=1

and the resulting power series have R as radius of convergence. If f (x), g(x) are analytic at x = x0 then so is f (x)g(x) and af +bg for any scalars a, b with radii of convergence at least that of the smaller of the radii of convergence the series for f (x), g(x). If f (x) is analytic at x = x0 and f (x0 ) 6= 0 then 1/f (x0 ) is analytic at x = x0 with radius of convergence equal to the distance from x0 to the nearest zero of f (x) in the complex plane. The following theorem shows that linear DE’s with analytic coefficients at x0 have analytic solutions at x0 with radius of convergence as big as the smallest of the radii of convergence of the coefficient functions.

1. 1.1

Series Solutions near a Ordinary Point Theorem

If p1 (x), p2 (x), . . . , pn (x), q(x) are analytic at x = x0 , the solutions of the DE y (n) + p1 (x)y (n−1) + · · · + pn (x)y = q(x) are analytic with radius of convergence ≥ the smallest of the radii of convergence of the coefficient functions p1 (x), p2 (x), . . . , pn (x), q(x). The proof of this result follows from the proof of fundamental existence and uniqueness theorem for linear DE’s using elementary properties of analytic functions and the fact that uniform limits of analytic functions are analytic. Example 1. The coefficients of the DE y 00 + y = 0 are analytic everywhere, in particular at x = 0. Any solution y = y(x) has therefore a series representation y=

∞ X

an xn

n=0

with infinite radius of convergence. We have y0 =

∞ X

(n + 1)an+1 xn ,

n=0

y 00 =

∞ X

(n + 1)(n + 2)an+2 xn .

n=0

Therefore, we have y 00 + y =

∞ X

((n + 1)(n + 2)an+2 + an )xn = 0

n=0

SERIES SOLUTION OF LINEARDIFFERENTIAL EQUATIONS

85

for all x. It follows that (n + 1)(n + 2)an+2 + an = 0 for n ≥ 0. Thus an an+2 = − , for n ≥ 0 (n + 1)(n + 2) from which we obtain a0 a1 a2 a0 a2 = − , a3 = − , a4 = − = , 1·2 2·3 3·4 1·2·3·4 a3 a1 a5 = − = . 4·5 2·3·4·5 By induction one obtains a0 a1 a2n = (−1)n , a2n+1 = (−1)n (2n)! (2n + 1)! and hence that y = a0

∞ X

(−1)n

n=0

∞ X x2n+1 x2n + a1 (−1)n = a0 cos(x) + a1 sin(x). (2n)! (2n + 1)! n=0

Example 2. The simplest non-constant DE is y 00 + xy = 0 which is known as Airy’s equation. Its coefficients are analytic everywhere and so the solutions have a series representation y=

∞ X

an xn

n=0

with infinite radius of convergence. We have y 00 + xy = =

∞ X

(n + 1)(n + 2)an+2 xn +

∞ X

n=0 ∞ X

n=0 ∞ X

n=0

n=1

(n + 1)(n + 2)an+2 xn +

= 2a2 +

∞ X

an xn+1 , an−1 xn ,

(4.1)

((n + 1)(n + 2)an+2 + an−1 )xn = 0

n=1

from which we get a2 = 0, (n + 1)(n + 2)an+2 + an−1 = 0 for n ≥ 1. Since a2 = 0 and an−1 , for n ≥ 1 an+2 = − (n + 1)(n + 2) we have a3 = −

a0 a1 , a4 = − , a5 = 0, 2·3 3·4 a3 a0 a6 = − = , 5·6 2·3·5·6 a4 a1 a7 = − = . 6·7 3·4·6·7

86

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

By induction we get a3n+2 = 0 for n ≥ 0 and a3n = (−1)n a3n+1 = (−1)n

a0 , 2 · 3 · 5 · 6 · · · (3n − 1) · 3n a1 . 3 · 4 · 6 · 7 · · · (3n) · (3n + 1)

Hence y = a0 y1 + a1 y2 with y1 = 1 −

x3 x6 x3n + − · · · + (−1)n +···, 2·3 2·3·5·6 2 · 3 · 5 · 6 · · · (3n − 1) · 3n

y2 = x −

x4 x7 x3n+1 + − · · · (−1)n +···. 3·4 3·4·6·7 3 · 4 · 6 · 7 · · · (3n) · (3n + 1)

For positive x the solutions of the DE y 00 + xy = 0 behave like the solutions to a mass-spring system with variable spring constant. The solutions oscillate for x > 0 with increasing frequency as |x| → ∞. For x < 0 the solutions are monotone. For example, y1 , y2 are increasing functions of x for x ≤ 0.

PART (II): SERIES SOLUTION NEAR A REGULAR SINGULAR POINT

In this lecture we investigate series solutions for the general linear DE a0 (x)y (n) + a1 (x)y (n−1) + · · · + an (x)y = b(x), where the functions a1 , a2 , . . . , an , b are analytic at x = x0 . If a0 (x0 ) 6= 0 the point x = x0 is called an ordinary point of the DE. In this case, the solutions are analytic at x = x0 since the normalized DE y (n) + p1 (x)y (n−1) + · · · + pn (x)y = q(x), where pi (x) = ai (x)/a0 (x), q(x) = b(x)/a0 (x), has coefficient functions which are analytic at x = x0 . If a0 (x0 ) = 0, the point x = x0 is said to be a singular point for the DE. If k is the multiplicity of the zero of a0 (x) at x = x0 and the multiplicities of the other coefficient functions at x = x0 is as big then, on cancelling the common factor (x − x0 )k for x 6= x0 , the DE obtained holds even for x = x0 by continuity, has analytic coefficient functions at x = x0 and x = x0 is an ordinary point. In this case the singularity is said to be removable. For example, the DE xy 00 + sin(x)y 0 + xy = 0 has a removable singularity at x = 0.

2.

Series Solutions near a Regular Singular Point

In general, the solution of a linear DE in a neighborhood of a singularity is extremely difficult. However, there is an important special case where this can be done. For simplicity, we treat the case of the general second order homogeneous DE a0 (x)y 00 + a1 (x)y 0 + a2 (x)y = 0,

(x > x0 ),

with a singular point at x = x0 . Without loss of generality we can, after possibly a change of variable x − x0 = t, assume that x0 = 0. We say

87

88

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

that x = 0 is a regular singular point if the normalized DE y 00 + p(x)y 0 + q(x)y = 0,

(x > 0),

is such that xp(x) and x2 q(x) are analytic at x = 0. A necessary and sufficient condition for this is that lim x2 q(x) = q0

lim xp(x) = p0 ,

x→0

x→0

exist and are finite. In this case xp(x) = p0 + p1 x + · · · + pn xn + · · · ,

x2 q(x) = q0 + q1 x + · · · + qn xn + · · ·

and the given DE has the same solutions as the DE x2 y 00 + x(xp(x))y 0 + x2 q(x)y = 0. This DE is an Euler DE if xp(x) = p0 , x2 q(x) = q0 . This suggests that we should look for solutions of the form r

y=x

Ã∞ X

!

n

an x

=

n=0

∞ X

an xn+r ,

n=0

with a0 6= 0. Substituting this in the DE gives ∞ X

n+r

(n + r)(n + r − 1)an x

n=0

+

Ã∞ X

pn x

n

!Ã ∞ X

n=0

+

Ã∞ X

!

(n + r)an x

n=0

n

qn x

!Ã ∞ X

n=0

n+r

! n+r

an x

=0

n=0

which, on expansion and simplification, becomes a0 F (r)xr +

∞ n X

F (n + r)an + [(n + r − 1)p1 + q1 ]an−1 + · · ·

n=1

o

+(rpn + qn )a0 xn+r = 0, (4.2) where F (r) = r(r − 1) + p0 r + q0 . Equating coefficients to zero, we get r(r − 1) + p0 r + q0 = 0,

(4.3)

the indicial equation, and F (n + r)an = −[(n + r − 1)p1 + q1 ]an−1 − · · · − (rpn + qn )a0 (4.4) for n ≥ 1. The indicial equation (4.3) has two roots: r1 , r2 . Three cases should be discussed separately.

89

SERIES SOLUTION OF LINEARDIFFERENTIAL EQUATIONS

2.1

Case (I): The roots (r1 − r2 6= N )

Two roots do’nt differ by an integer. In this case, the above recursive equation (4.4) determines an uniquely for r = r1 and r = r2 . If an (ri ) is the solution for r = ri and a0 = 1, we obtain the linearly independent solutions y1 = x

r1

Ã∞ X

!

an (r1 )x

n

r2

,

y2 = x

Ã∞ X

n=0

!

n

an (r2 )x

.

n=0

It can be shown that the radius of convergence of the infinite series is the distance to the singularity of the DE nearest to the singularity x = 0. If r1 − r2 = N ≥ 0, the above recursion equations can be solved for r = r1 as above to give a solution y1 = xr1

Ã∞ X

!

an (r1 )xn .

n=0

A second linearly independent solution can then be found by reduction of order. However, the series calculations can be quite involved and a simpler method exists which is based on solving the recursion equation for an (r) as a ratio of polynomials of r. This can always be done since F (n + r) is not the zero polynomial for any n ≥ 0. If an (r) is the solution with a0 (r) = 1 and we let r

y = y(x, r) = x

Ã∞ X

!

n

an (r)x

.

(4.5)

n=0

Thus, we have the following equality with two variables (x, r): x2 y 00 + x2 p(x)y 0 + x2 q(x)y = a0 F (r)xr = (r − r1 )(r − r2 )xr . (4.6)

2.2

Case (II): The roots (r1 = r2 )

In this case, from the equality (4.6) we get x2 y 00 + x2 p(x)y 0 + x2 q(x)y = (r − r1 )2 xr . Differentiating this equation with respect to r, we get µ

x2

∂y ∂r

¶00

µ

+ x2 p(x)

∂y ∂r

¶0

+ x2 q(x)

Setting r = r1 , we find that ∂y y2 = (x, r1 ) = xr1 ∂r

Ã∞ X n=0

∂y = 2(r − r1 ) + (r − r1 )2 xr ln(x). ∂r !

an (r1 )xn ln(x) + xr1

∞ X n=0

a0n (r1 )xn

90

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

= y1 ln(x) + xr1

∞ X

a0n (r1 )xn ,

n=0

a0n (r)

where is the derivative of an (r) with respect to r. This is a second linearly independent solution. Since this solution is unbounded as x → 0, any solution of the given DE which is bounded as x → 0 must be a scalar multiple of y1 .

2.3

Case (III): The roots (r1 − r2 = N > 0)

For this case, we let z(x, r) = (r − r2 )y(x, r). Thus, from the equality (4.6) we get x2 z 00 + x2 p(x)z 0 + x2 q(x)z = (r − r1 )(r − r2 )2 xr . Differentiating this equation with respect to r, we get µ

x

2

∂z ∂r

¶00

µ

∂z + x p(x) ∂r 2

¶0

+ x2 q(x)

∂z = (r − r1 )(r − r2 )2 xr ln(x) ∂r h

i

+(r − r2 ) (r − r2 ) + 2(r − r1 ) xr . Setting r = r2 , we see that y2 = ∂z ∂r (x, r2 ) is a solution of the given DE. Letting bn (r) = (r − r2 )an (r), we have h

i

F (n + r)bn (r) = − (n + r − 1)p1 + q1 bn−1 (r) − · · · −(rpn + qn )b0 (r) and

Ã

y2 = lim

r

r→r2

x ln(x)

∞ X n=0

n

(4.7)

r

bn (r)x + x

∞ X

!

b0n (r)xn

.

(4.8)

n=0

Note that an (r2 ) 6= 0, for n = 1, 2, . . . N − 1. Hence, we have b0 (r2 ) = b1 (r2 ) = b2 (r2 ) = · · · = bN −1 (r2 ) = 0. However, aN (r2 ) = ∞, as F (r2 + N ) = F (r1 ) = 0. Hence, we have bN (r2 ) = lim (r − r2 )an (r) = a < ∞, r→r2

subsequently,

lim xr ln(x)bN (r)xN = axr1 ln(x).

r→r2

Furthermore, F (N + 1 + r2 )bN +1 (r2 ) = F (1 + r1 )bN +1 (r2 ) = −(r1 p1 + q1 )bN (r2 ) − · · · − (r2 pN +1 + qN +1 )b0 (r2 ) = −(r1 p1 + q1 )bN (r2 ) (4.9)

SERIES SOLUTION OF LINEARDIFFERENTIAL EQUATIONS

91

Thus, bN +1 (r2 ) =

(r1 p1 + q1 ) (r1 p1 + q1 ) bN (r2 ) = a = aa1 (r1 ). F (1 + r1 ) F (1 + r1 )

(4.10)

Similarly, we have F (N + 2 + r2 )bN +2 (r2 ) = F (2 + r1 )bN +2 (r2 ) = −[(1 + r1 )p1 + q1 ]bN +1 (r2 ) − (r1 p2 + q2 )bN (r2 ) − · · · − (r2 pN +2 + qN +2 )b0 (r2 ) = −a[(1 + r1 )p1 + q1 ]a1 (r1 ) − a(r1 p2 + q2 ), (4.11) then we obtain bN +2 (r2 ) = −a

[(1 + r1 )p1 + q1 ]a1 (r1 ) + (r1 p2 + q2 ) = aa2 (r1 ).(4.12) F (2 + r1 )

In general, we can write bN +k (r2 ) = aak (r1 ).

(4.13)

Substituting the above results to (4.8), we finally derive r1

y2 = ax

Ã∞ X

! n

an (r1 )x

n=0

= ay1 ln(x) + xr2

Ã∞ X

r2

ln(x) + x

!

Ã∞ X

!

b0n (r2 )xn

n=0

b0n (r2 )xn .

(4.14)

n=0

This gives a second linearly independent solution. The above method is due to Frobenius and is called the Frobenius method. Example 1. The DE 2xy 00 + y 0 + 2xy = 0 has a regular singular point at x = 0 since xp(x) = 1/2 and x2 q(x) = x2 . The indicial equation is 1 1 r(r − 1) + r = r(r − ). 2 2 The roots are r1 = 1/2, r2 = 0 which do not differ by an integer. We have (r + 1)(r + 12 )a1 = 0, (n + r)(n + r − 12 )an = −an−2

for n ≥ 2,

(4.15)

92

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

so that an = −2an−2 /(r + n)(2r + 2n − 1) for n ≥ 2. Hence 0 = a1 = a3 = · · · a2n+1 for n ≥ 0 and 2 a0 , (r + 2)(2r + 3) 2 22 a4 = − a2 = a0 . (r + 4)(2r + 7) (r + 2)(r + 4)(2r + 3)(2r + 7) a2 = −

It follows by induction that 2n (r + 2)(r + 4) · · · (r + 2n) 1 × a0 . (2r + 3)(2r + 4) · · · (2r + 2n − 1)

a2n = (−1)n

(4.16)

Setting, r = 1/2, 0, a0 = 1, we get y1 =

∞ √ X x

x2n , (5 · 9 · · · (4n + 1))n! n=0

y2 =

∞ X

x2n . (3 · 7 · · · · (4n − 1))n! n=0

The infinite series have an infinite radius of convergence since x = 0 is the only singular point of the DE. Example 2. The DE xy 00 + y 0 + y = 0 has a regular singular point at x = 0 with xp(x) = 1, x2 q(x) = x. The indicial equation is r(r − 1) + r = r2 = 0. This equation has only one root x = 0. The recursion equation is (n + r)2 an = −an−1 ,

n ≥ 1.

The solution with a0 = 1 is an (r) = (−1)n

(r +

1)2 (r

1 . + 2)2 · · · (r + n)2

setting r = 0 gives the solution y1 =

∞ X

(−1)n

n=0

xn . (n!)2

Taking the derivative of an (r) with respect to r we get, using a0n (r) = an (r)

d ln [an (r)] dr

SERIES SOLUTION OF LINEARDIFFERENTIAL EQUATIONS

93

(logarithmic differentiation),we get µ

a0n (r) = −

¶

2 2 2 + + ··· + an (r) r+1 r+2 r+n

so that

1

a0n (0) = 2(−1)n 1

+

1 2

+ ··· + (n!)2

1 n

.

Therefore a second linearly independent solution is y2 = y1 ln(x) + 2

∞ X

1

(−1)n 1

n=1

+

1 2

+ ··· + (n!)2

1 n

xn .

The above series converge for all x. Any bounded solution of the given DE must be a scalar multiple of y1 .

PART (III): BESSEL FUNCTIONS

3.

Bessel Equation

In this lecture we study an important class of functions which are defined by the differential equation x2 y 00 + xy 0 + (x2 − ν 2 )y = 0, where ν ≥ 0 is a fixed parameter. This DE is known Bessel’s equation of order ν. This equation has x = 0 as its only singular point. Moreover, this singular point is a regular singular point since xp(x) = 1,

x2 q(x) = x2 − ν 2 .

Bessel’s equation can also be written y 00 +

1 0 ν2 y + (1 − 2 ) = 0 x x

which for x large is approximately the DE y 00 + y = 0 so that we can expect the solutions to oscillate for x large. The indicial equation is r(r − 1) + r − ν 2 = r − ν 2 whose roots are r1 = ν and r2 = −ν. The recursion equations are [(1 + r)2 − ν 2 ]a1 = 0,

[(n + r)2 − ν 2 ]an = −an−2 ,

for n ≥ 2.

The general solution of these equations is a2n+1 = 0 for n ≥ 0 and (−1)n a0 (r + 2 − ν)(r + 4 − ν) · · · (r + 2n − ν) 1 × . (r + 2 + ν)(r + 4 + ν) · · · (r + 2n + ν)

a2n (r) =

95

96

4.

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

The Case of Non-integer ν

If ν is not an integer and ν 6= 1/2, we have the case (I). Two linearly independent solutions of Bessel’s equation Jν (x), J−ν (x) can be obtained by taking r = ±ν, a0 = 1/2ν Γ(ν + 1). Since, in this case, a2n =

(−1)n a0 , 22n n!(r + 1)(r + 2) · · · (r + n)

we have for r = ±ν ∞ X

(−1)n Jr (x) = n!Γ(r + n + 1) n=0

µ ¶2n+r

x 2

.

Recall that the Gamma function Γ(x) is defined for x ≥ −1 by Γ(x + 1) =

Z ∞ 0

e−t tx dt.

For x ≥ 0 we have Γ(x + 1) = xΓ(x), so that Γ(n + 1) = n! for n an integer ≥ 0. We have µ ¶

1 Γ 2

=

Z ∞ 0

−t −1/2

e t

dt = 2

Z ∞ 0

2

e−x dt =

√ π.

The Gamma function can be extended uniquely for all x except for x = 0, −1, −2, . . . , −n, . . . to a function which satisfies the identity Γ(x) = Γ(x)/x. This is true even if x is taken to be complex. The resulting function is analytic except at zero and the negative integers where it has a simple pole. These functions are called Bessel functions of first kind of order ν. As an exercise the reader can show that r

J 1 (x) = 2

5.

2 sin(x), πx

r

J− 1 = 2

2 cos(x). πx

The Case of ν = −m with m an integer ≥ 0

For this case, the first solution Jm (x) can be obtained as in the last section. As examples, we give some such solutions as follows: The Case of m = 0: J0 (x) =

∞ X (−1)n

22n (n!)2 n=0

x2n

SERIES SOLUTION OF LINEARDIFFERENTIAL EQUATIONS

97

The case m = 1: ∞ 1 xX (−1)n J1 (x) = y1 (x) = x2n . 2 2 n=0 22n n!(n + 1)!

To derive the second solution, one has to proceed differently. For ν = 0 the indicial equation has a repeated root, we have the case (II). One has a second solution of the form y2 = J0 (x) ln(x) +

∞ X

a02n (0)x2n

n=0

where a2n (r) =

(−1)n . (r + 2)2 (r + 4)2 · · · (r + 2n)2

It follows that

µ

a02n (r) 1 1 1 = −2 + + ··· a2n r+2 r+4 r + 2n so that

¶

µ

a02n (0) = − 1 + where we have defined

¶

1 1 + ··· + a2n (0) = −hn a2n (0), 2 n µ

¶

1 1 hn = 1 + + · · · + . 2 n Hence y2 = J0 (x) ln(x) +

∞ X (−1)n+1 hn 2n x . 2n 2

n=0

2 (n!)

Instead of y2 , the second solution is usually taken to be a certain linear combination of y2 and J0 . For example, the function Y0 (x) =

i 2h y2 (x) + (γ − ln 2)J0 (x) , π

where γ = lim (hn − ln n) ≈ 0.5772, is known as the Weber function n→∞ of order 0. The constant γ is known as Euler’s constant; it is not known whether γ is rational or not. If ν = −m, with m > 0, the the roots of the indicial equation differ by an integer, we have the case (III). Then one has a solution of the form y2 = aJm (x) ln(x) +

∞ X n=0

b02n (−m)x2n+m

98

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

where b2n (r) = (r + m)a2n (r) and a = b2m (−m). In the case m = 1 we have a0 = 1, a0 a = b2 (−1) = − , 2 b0 (r) = (r − r2 )a0 and for n ≥ 1, b2n (r) =

(−1)n a0 . (r + 3)(r + 5) · · · (r + 2n − 1)(r + 3)(r + 5) · · · (r + 2n + 1)

Subsequently, we have

b00 (r) = a0

and for n ≥ 1, µ

b02n (r)

=−

1 1 1 1 + + ··· + + r+3 r+5 r + 2n − 1 r + 3 ¶ 1 1 + ··· + b2n (r). + r+5 r + 2n + 1

From here, we obtain b00 (−1) = a0 b02n (−1) = −1 2 (hn + hn−1 )b2n (−1)

(4.17)

(n ≥ 1),

where b2n (−1) =

(−1)n a0 . 22n (n − 1)!n!

So that "

∞ X −1 1 (−1)n+1 (hn + hn−1 ) 2n y2 = y1 (x) ln(x) + 1+ x 2 x 22n+1 (n − 1)!n! n=1

"

∞ X (−1)n+1 (hn + hn−1 ) 2n 1 1+ x = −J1 (x) ln(x) + x 22n+1 (n − 1)!n! n=1

#

#

where, by convention, h0 = 0, (−1)! = 1. The Weber function of order 1 is defined to be Y1 (x) =

i 4h − y2 (x) + (γ − ln 2)J1 (x) . π

The case m > 1 is slightly more complicated and will not be treated here.

SERIES SOLUTION OF LINEARDIFFERENTIAL EQUATIONS

99

The second solutions y2 (x) of Bessel’s equation of order m ≥ 0 are unbounded as x → 0. It follows that any solution of Bessel’s equation of order m ≥ 0 which is bounded as x → 0 is a scalar multiple of Jm . The solutions which are unbounded as x → 0 are called Bessel functions of the second kind. The Weber functions are Bessel functions of the second kind.

Chapter 5 LAPLACE TRANSFORMS

101

PART (I): LAPLACE TRANSFORM AND ITS INVERSE

1.

Introduction

We begin our study of the Laplace Transform with a motivating example. This example illustrates the type of problem that the Laplace transform was designed to solve. A mass-spring system consisting of a single steel ball is suspended from the ceiling by a spring. For simplicity, we assume that the mass and spring constant are 1. Below the ball we introduce an electromagnet controlled by a switch. Assume that, we on, the electromagnet exerts a unit force on the ball. After the ball is in equilibrium for 10 seconds the electromagnet is turned on for 2π seconds and then turned off. Let y = y(t) be the downward displacement of the ball from the equilibrium position at time t. To describe the motion of the ball using techniques previously developed we have to divide the problem into three parts: (I) 0 ≤ t < 10; (II) 10 ≤ t < 10 + 2π; (III) 10 + 2π ≤ t. The initial value problem determining the motion in part I is y 00 + y = 0,

y(0) = y 0 (0) = 0.

The solution is y(t) = 0, 0 ≤ t < 10. Taking limits as t → 10 from the left, we find y(10) = y 0 (10) = 0. The initial value problem determining the motion in part II is y 00 + y = 1,

y(10) = y 0 (10) = 0.

The solution is y(t) = 1 − cos(t − 10), 10 ≤ t < 2π + 10. Taking limits as t → 10 + 2π from the left, we get y(10 + 2π) = y 0 (10 + 2π) = 0. The initial value problem for the last part is y 00 + y = 0,

y(10 + 2π) = y 0 (10 + 2π) = 0

103

104

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

which has the solution y(t) = 0, 10 + 2π ≤ t. Putting all this together, we have 0, 0 ≤ t < 10, 1 − cos(t − 10), 10 ≤ t < 10 + 2π, y(t) = 0, 10 + 2π ≤ t. The function y(t) is continuous with continuous derivative y 0 (t) =

0,

0 ≤ t < 10, sin(t − 10), 10 ≤ t < 10 + 2π, 0, 10 + 2π ≤ t.

However the function y 0 (t) is not differentiable at t = 10 and t = 10+2π. In fact 0, 0 ≤ t < 10, 00 cos(t − 10), 10 < t < 10 + 2π, y (t) = 0, 10 + 2π < t. The left hand and right hand limits of f 00 (t) at t = 10 are 0 and 1 respectively. At t = 10+2π they are 1 and 0. If we extend y 00 (t) by using the left-hand or righthand limits at 10 and 10+2π the resulting function is not continuous. Such a function with only jump discontinuities is said to be piecewise continuous. If we try to write the differential equation of the system we have y 00 + y = f (t) =

0,

1, 0,

0 ≤ t < 10, 10 ≤ t < 10 + 2π, 10 + 2π ≤ t.

Here f (t) is piecewise continuous and any solution would also have y 00 piecewise continuous. By a solution we mean any function y = y(t) satisfying the DE for those t not equal to the points of discontinuity of f (t). In this case we have shown that a solution exists with y(t), y 0 (t) continuous. In the same way, one can show that in general such solutions exist using the fundamental theorem. What we want to describe now is a mechanism for doing such problems without having to divide the problem into parts. This mechanism is the Laplace transform.

2. 2.1

Laplace Transform Definition

Let f (t) be a function defined for t ≥ 0. The function f (t) is said to be piecewise continuous if (1) f (t) converges to a finite limit f (0+ ) as t → 0+

105

LAPLACE TRANSFORMS

(2) for any c > 0, the left and right hand limits f (c− ), f (c+ ) of f (t) at c exist and are finite. (3) f (c− ) = f (c+ ) = f (c) for every c > 0 except possibly a finite set of points or an infinite sequence of points converging to +∞. Thus the only points of discontinuity of f (t) are jump discontinuities. The function is said to be normalized if f (c) = f (c+ ) for every c ≥ 0. The Laplace transform F (s) = L{f (t)} is the function of a new variable s defined by F (s) =

Z ∞ 0

e

−st

f (t)dt =

lim

Z N

N →+∞ 0

e−st f (t)dt.

An important class of functions for which the integral converges are the functions of exponential order. The function f (t) is said to be of exponential order if there are constants a, M such that |f (t)| ≤ M eat for all t. the solutions of constant coefficient homogeneous DE’s are all of exponential order. The convergence of the improper integral follows from Z N Z N 1 e−(s−a) −st |e f (t)|dt ≤ M e−(s−a)t dt = − s−a s−a 0 0 which shows that the improper integral converges absolutely when s > a. It shows that F (s) → 0 as s → ∞. The calculation also shows that 1 L{eat } = s−a for s > a. Setting a = 0, we get L{1} =

2.2

1 s

for s > 0.

Basic Properties and Formulas

The above holds when f (t) is complex valued and s = σ + iτ is complex. The integral exists in this case for σ > a. For example, this yields 1 1 , L{e−it } = . L{eit } = s−i s+i

2.2.1

Linearity of the transform

L{af (t) + bf (t)} = aL{f (t)} + bL{g(t)}. Using this linearity property of the Laplace transform and using sin(t) = it (e − e−it )/2i, cos(t) = (eit + e−it )/2, we find 1 L{sin(bt)} = 2i

µ

1 1 − s − bi s + bi

¶

=

s2

b , + b2

106

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

1 L{cos(bt)} = 2

µ

1 1 + s − bi s + bi

¶

=

s2

s , + b2

for s > 0.

2.2.2 Formula (I) The following two identities follow from the definition of the Laplace transform after a change of variable. L{eat f (t)}(s) = L{f (t)}(s − a), 1 L{f (bt)}(s) = L{f (t)}(s/b). b Using the first of these formulas, we get L{eat sin(bt)} =

b , (s − a)2 + b2

L{eat cos(t)} =

s−a . (s − a)2 + b2

2.2.3 Formula (II) The next formula will allow us to find the Laplace transform for all the functions that are annihilated by a constant coefficient differential operator. dn L{tn f (t)}(s) = (−1)n n L{f (t)}(s). ds For n = 1 this follows from the definition of the Laplace transform on differentiating with respect s and taking the derivative inside the integral. The general case follows by induction. For example, using this formula, we obtain using f (t) = 1 L{tn }(s) = −

dn 1 n! = n+1 . n ds s s

With f (t) = sin(t) and f (t) = cos(t) we get L{t sin(bt)}(s) = − L{t cos(bt)}(s) = −

d b 2bs = 2 , 2 2 ds s + b (s + b2 )2

s s2 − b2 1 2b2 d = = − . ds s2 + b2 (s2 + b2 )2 s2 + b2 (s2 + b2 )2

2.2.4 Formula (III) The next formula shows how to compute the Laplace transform of f 0 (t) in terms of the Laplace transform of f (t). L{f 0 (t)}(s) = sL{f (t)}(s) − f (0).

107

LAPLACE TRANSFORMS

This follows from L{f 0 (t)}(s) =

Z ∞ 0

=s

¯∞ ¯

e−st f 0 (t)dt = e−st f (t)¯

Z ∞ 0

0

+s

Z ∞ 0

e−st f (t)dt

e−st f (t)dt − f (0)

(5.1)

since e−st f (t) converges to 0 as t → +∞ in the domain of definition of the Laplace transform of f (t). To ensure that the first integral is defined, we have to assume f 0 (t) is piecewise continuous. Repeated applications of this formula give L{f (n) (t)}(s) = sn L{f (t)}(s) − sn−1 f (0) − sn−2 f 0 (0) − · · · − f n−1 (0). The following theorem is important for the application of the Laplace transform to differential equations.

3. 3.1

Inverse Laplace Transform Theorem:

If f (t), g(t) are normalized piecewise continuous functions of exponential order then L{f (t)} = L{g(t)} implies f = g.

3.2

Definition

If F (s) is the Laplace of the normalized piecewise continuous function f (t) of exponential order then f (t) is called the inverse Laplace transform of F (s). This is denoted by F (s) = L−1 {f (t)}. Note that the inverse Laplace transform is also linear. Using the Laplace transforms we found for t sin(bt), t cos(bt) we find ½

L−1 and

½

L−1

s 2 (s + b2 )2

1 (s2 + b2 )2

¾

=

¾

=

1 t sin(bt), 2b

1 1 sin(bt) − 2 t cos(bt). 2b3 2b

PART (II): SOLVE DE’S WITH LAPLACE TRANSFORMS

4.

Solve IVP of DE’s with Laplace Transform Method

In this lecture we will, by using examples, show how to use Laplace transforms in solving differential equations with constant coefficients.

4.1

Example 1

Consider the initial value problem y 00 + y 0 + y = sin(t),

y(0) = 1, y 0 (0) = −1.

Step 1 Let Y (s) = L{y(t)}, we have L{y 0 (t)} = sY (s) − y(0) = sY (s) − 1, L{y 00 (t)} = s2 Y (s) − sy(0) − y 0 (0) = s2 Y (s) − s + 1. Taking Laplace transforms of the DE, we get (s2 + s + 1)Y (s) − s =

s2

1 . +1

Step 2 Solving for Y (s), we get Y (s) =

s 1 + . s2 + s + 1 (s2 + s + 1)(s2 + 1)

109

110

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

Step 3 Finding the inverse laplace transform. ½

y(t) = L−1 {Y (s)} = L−1

s s2 + s + 1

¾

½

¾

1 . (s2 + s + 1)(s2 + 1)

+ L−1

Since s2

s s s + 1/2 √ = = 2 +s+1 (s + 1/2) + 3/4 (s + 1/2)2 + ( 3/2)2 √ 1 3/2 √ −√ 2 3 (s + 1/2) + ( 3/2)2

we have ½

L

−1

s 2 s +s+1

¾

√ √ 1 = e−t/2 cos( 3 t/2) − √ e−t/2 sin( 3 t/2). 3

Using partial fractions we have 1 As + B Cs + D = 2 + 2 . (s2 + s + 1)(s2 + 1) s +s+1 s +1 Multiplying both sides by (s2 + 1)(s2 + s + 1) and collecting terms, we find 1 = (A + C)s3 + (B + C + D)s2 + (A + C + D)s + B + D. Equating coefficients, we get A + C = 0, B + C + D = 0, A + C + D = 0, B + D = 1, from which we get A = B = 1, C = −1, D = 0, so that ½

L

−1

1 2 (s + s + 1)(s2 + 1)

¾

½

¾

s =L + L−1 2 s +s+1 ½ ¾ s −1 −L . s2 + 1 −1

½

1 2 s +s+1

¾

Since ½

L

−1

1 2 s +s+1

we obtain

¾

√ 2 = √ e−t/2 sin( 3 t/2), 3

½ −1

L

s 2 s +1

√ y(t) = 2e−t/2 cos( 3 t/2) − cos(t).

¾

= cos(t)

111

LAPLACE TRANSFORMS

4.2

Example 2

As we have known, a higher order DE can be reduced to a system of DE’s. Let us consider the system dx = −2x + y, dt dy = x − 2y dt

(5.2)

with the initial conditions x(0) = 1, y(0) = 2.

Step 1 Taking Laplace transforms the system becomes sX(s) − 1 = −2X(s) + Y (s), sY (s) − 2 = X(s) − 2Y (s),

(5.3)

where X(s) = L{x(t)}, Y (s) = L{y(t)}.

Step 2 Solving for X(s), Y (s). The above linear system of equations can be written in the form: (s + 2)X(s) − Y (s) = 1, −X(s) + (s + 2)Y (s) = 2.

(5.4)

The determinant of the coefficient matrix is s2 + 4s + 3 = (s + 1)(s + 3). Using Cramer’s rule we get X(s) =

s2

s+4 , + 4s + 3

Y (s) =

s2

2s + 5 . + 4s + 3

Step 3 Finding the inverse Laplace transform. Since s+4 3/2 1/2 = − , (s + 1)(s + 3) s+1 s+3

2s + 5 3/2 1/2 = + , (s + 1)(s + 3) s+1 s+3

we obtain 3 1 x(t) = e−t − e−3t , 2 2

3 1 y(t) = e−t + e−3t . 2 2

The Laplace transform reduces the solution of differential equations to a partial fractions calculation. If F (s) = P (s)/Q(s) is a ratio of

112

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

polynomials with the degree of P (s) less than the degree of Q(s) then F (s) can be written as a sum of terms each of which corresponds to an irreducible factor of Q(s). Each factor Q(s) of the form s − a contributes the terms Ar A1 A1 + ··· + + 2 s − a (s − a) (s − a)r where r is the multiplicity of the factor s − a. Each irreducible quadratic factor s2 + as + b contributes the terms A1 s + B1 A2 s + B2 Ar s + Br + 2 + ··· + 2 2 2 s + as + b (s + as + b) (s + as + b)r where r is the degree of multiplicity of the factor s2 + as + b.

PART (III): FURTHER STUDIES OF LAPLACE TRANSFORM

5. 5.1

Step Function Definition ½

uc (t) =

5.2

0 1

t < c, t ≥ c.

Laplace transform of unit step function L{uc (t)} =

e−cs . s

One can derive L{uc (t)f (t − c)} = e−cs F (s).

6. 6.1

Impulse Function Definition

Let dτ (t) = It follows that I(τ ) =

½ 1

2τ

0

Z ∞ −∞

|t| < τ, |t| ≥ τ.

dτ (t)dt = 1.

Now, consider the limit, ½

δ(t) = lim dτ (t) = τ →0

113

0 ∞

t 6= 0, t = 0,

114

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

which is called the Dirac δ-function. Evidently, the Dirac δ-function has the following properties: 1

Z ∞ −∞

2

Z B A

3

Z B A

6.2

δ(t)dt = 1. ½

δ(t − c)dt =

0 1

½

δ(t − c)f (t)dt =

¯ (A, B), c∈ c ∈ (A, B).

0 f (c)

¯ (A, B), c∈ c ∈ (A, B).

Laplace transform of unit step function L{δ(t − c)} =

½ −cs e

0

c > 0, c < 0.

One can derive L{δ(t − c)f (t)} = e−cs f (c),

7. 7.1

(c > 0).

Convolution Integral Theorem

Given L{f (t)} = F (s), one can derive

L{g(t)} = G(s),

L−1 {F (s) · G(s)} = f (t) ∗ g(t),

where f (t) ∗ g(t) =

Z τ

is called the convolution integral.

0

f (t − τ )g(τ )dτ

Chapter 6 (*) SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS

115

(*) PART (I): INTRODUCTION OF SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS

In this and the following lecture we will give an introduction to systems of differential equations. For simplicity, we will limit ourselves to systems of two equations with two unknowns. The techniques introduced can be used to solve systems with more equations and unknowns. As a motivational example, consider the the following problem.

1.

Mathematical Formulation of a Practical Problem

Two large tanks, each holding 24 liters of brine, are interconnected by two pipes. Fresh water flows into tank A a the rate of 6 L/min, and fluid is drained out tank B at the same rate. Also, 8 L/min of fluid are pumped from tank A to tank B and 2 L/min from tank B to tank A. The solutions in each tank are well stirred sot that they are homogeneous. If, initially, tank A contains 5 in solution and Tank B contains 2 kg, find the mass of salt in the tanks at any time t. To solve this problem, let x(t) and y(t) be the mass of salt in tanks A and B respectively. The variables x, y satisfy the system dx 1 = −1 3 x + 12 y, dt (6.1) dy = 13 x − 13 y. dt The first equation gives y = 12 dx dt + 4x. Substituting this in the second equation and simplifying, we get 1 d2 x 2 dx + + x = 0. 2 dt 3 dt 12

117

118

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

The general solution of this DE is x = c1 e−t/2 + c2 e−t/6 . −t/2 + 2c e−t/6 . Thus the general This gives y = 12 dx 2 dt + 4x = −2c1 e solution of the system is

x = c1 e−t/2 + c2 e−t/6 , y = −2c1 e−t/2 + 2c2 e−t/6 .

(6.2)

These equations can be written in matrix form as µ

X=

x y

¶

µ

= c1 e

−t/2

1 −2

µ ¶

¶

+ c2 e

−t/6

1 2

.

Using the initial condition x(0) = 5, y(0) = 2, we find c1 = 2, c2 = 3. Geometrically, these equations are the parametric equations of a curve (trajectory of the DE) in the xy-plane (phase plane of the DE). As t → ∞ we have (x(t), y(t)) → (0, 0). The constant solution x(t) = y(t) = 0 is called an equilibrium solution of our system. This solution is said to be asymptotically stable if the general solution converges to it as t → ∞. A system is called stable if the trajectories are all bounded as t → ∞. Our system can be written in matrix form as dX dt = AX where µ

A=

−1/3 1/12 1/3 −1/3

¶

X.

The 2 × 2 matrix A is called the matrix of the system. The polynomial 2 1 r2 − tr(A)r + det(A) = r2 + r + 3 12 where tr(A) is the trace of A (sum of diagonal entries) and det(A) is the determinant of A is called the characteristic polynomial of A. Notice that this polynomial is the characteristic polynomial of the differential equation for x. The equations µ

A µ

¶

1 −2

¶

−1 = 2

µ ¶

µ

1 −2

¶

µ ¶

,

A

1 2

−1 = 6

µ ¶

1 2

1 1 identify and as eigenvectors of A with eigenvalues −1/2 −2 2 and −1/6 respectively. More generally, a non-zero column vector X is an eigenvector of a square matrix A with eigenvalue r if AX = rX or , equivalently, (rI − A)X = 0. The latter is a homogeneous system

(*) SYSTEMS OF LINEAR DIFFERENTIALEQUATIONS

119

of linear equations with coefficient matrix rI − A. Such a system has a non-zero solution if and only if det(rI − A) = 0. Notice that det(rI − A) = r2 − (a + d)r + ad − bc is the characteristic polynomial of A. If, in the above mixing problem, brine at a concentration of 1/2 kg/L was pumped into tank A instead of pure water the system would be dx 1 = −1 3 x + 12 y + 3, dt dy = 13 x − 13 y, dt

(6.3)

a non-homogeneous system. Here an equilibrium solution would be x(t) = a, y(t) = b where (a, b) was a solution of −1 1 3 x + 12 y 1 1 3x − 3y

= −3, = 0.

(6.4)

In this case a = b = 12. The variables x∗ = x − 12, y ∗ = y − 12 then satisfy the homogeneous system dx∗ 1 ∗ ∗ = −1 3 x + 12 y , dt∗ dy = 13 x∗ − 31 y ∗ . dt

(6.5)

Solving this system as above for x∗ , y ∗ we get x = x∗ + 12, y = y ∗ + 12 as the general solution for x, y.

2.

(2 × 2) System of Linear Equations

We now describe the solution of the system dX dt = AX for an arbitrary 2 × 2 matrix A. In practice, one can use the elimination method or the eigenvector method but we shall use the eigenvector method as it gives an explicit description of the solution. There are three main cases depending on whether the discriminant ∆ = tr(A)2 − 4 det(A) of the characteristic polynomial of A is > 0, < 0, = 0.

2.1

Case 1: ∆ > 0

In this case the roots r1 , r2 of the characteristic polynomial are real and unequal, say r1 < r2 . Let Pi be an eigenvector with eigenvalue ri .

120

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

Then P1 is not a scalar multiple of P2 and so the matrix P with columns P1 , P2 is invertible. After possibly replacing P2 by −P2 , we can assume that det(P ) > 0. The equation µ

AP = P shows that

r1 0 0 r2

µ

P

−1

AP =

r1 0 0 r2

¶

¶

. µ

If we make the change of variable X = P U with U =

¶

u , our system v

becomes

dU dU = AP U or = P −1 AP U. dt dt Hence, our system reduces to the uncoupled system P

du = r1 u, dt

dv = r2 v dt

which has the general solution u = c1 er1 t , v = c2 er2 t . Thus the general solution of the given system is X = P U = uP1 + vP2 = c1 er1 t P1 + c2 er2 t P2 . Since tr(A) = r1 + r2 , det(A) = r1 r2 , we see that x(t), y(t) = (0, 0) is an asymptotically stable equilibrium solution if and only if tr(A) < 0 and det(A) > 0. The system is unstable if det(A) < 0 or det(A) ≥ 0 and tr(A) ≥ 0.

2.2

Case 2: ∆ < 0

In this case the roots of the characteristic polynomial are complex numbers q r = α ± iω = tr(A)/2 ± i ∆/4. The corresponding eigenvectors of A are (complex) scalar multiples of µ

1 σ ± iτ

¶

where σ = (α − a)/b, τ = ω/b. If X is a real solution we must have X = V + V with 1 V = (c1 + ic2 )eαt (cos(ωt) + i sin(ωt)) 2

µ

1 σ + iτ

¶

.

121

(*) SYSTEMS OF LINEAR DIFFERENTIALEQUATIONS

It follows that µ αt

X = e (c1 cos(ωt) − c2 sin(ωt))

1 σ

¶

µ αt

+ e (c1 sin(ωt) + c2 cos(ωt))

0 τ

¶

.

The trajectories are spirals if tr(A) 6= 0 and ellipses if tr(A) = 0. The system is asymptotically stable if tr(A) < 0 and unstable if tr(A) > 0.

2.3

Case 3: ∆ = 0

Here the characteristic polynomial has only one root r. If A = rI the system is dx dy = rx, = ry. dt dt which has the general solution x = c1 ert , y = c2 ert . Thus the system is asymptotically stable if tr(A) < 0, stable if tr(A) = 0 and unstable if tr(A) > 0. Now suppose A 6= rI. If P1 is an eigenvector with eigenvalue r and P2 is chosen with (A − rI)P1 6= 0, the matrix P with columns P1 , P2 is invertible and µ ¶ r 1 −1 P AP = . 0r Setting as before X = P U we get the system du = ru + v, dt

dv = rv dt

which has the general solution u = c1 ert + c2 tert , v = c2 ert . Hence the given system has the general solution X = uP1 + vP2 = (c1 ert + c2 tert )P1 + c2 ert P2 . The trajectories are asymptotically stable if tr(A) < 0 and unstable if tr(A) ≥ 0. A non-homogeneous system dX dt = AX + B having an equilibrium solution x(t) = x1 , y(t) = y1 can be solved by introducing new variables x∗ = x − x1 , y ∗ = y − y1 . Since AX ∗ + B = 0 we have dX ∗ = AX ∗ , dt a homogeneous system which can be solved as above.

(*) PART (II): EIGENVECTOR METHOD

In this lecture we will apply the eigenvector method to the solution of a second order system of the type arising in the solution of a mass-spring system with two masses. The system we will consider consists of two masses with mass m1 , m2 connected by a spring with spring constant k2 . The first mass is attached to the ceiling of a room by a spring with spring constant k1 and the second mass is attached to the floor by a spring with spring constant k3 at a point immediately below the point of attachment to the ceiling. Assume that the system is under tension and in equilibrium. If x1 (t), x1 (t) are the displacements of the two masses from their equilibrium position at time t, the positive direction being upward, then the motion of the system is determined by the system d2 x1 = −k1 x1 − k2 (x1 − x2 ) = −(k1 + k2 )x1 + k2 x2 , 2 dt 2 d x2 m2 2 = k2 (x1 − x2 ) − k3 x2 = k2 x1 − (k2 + k3 )x2 . dt

m1

The system can be written in matrix form µ

X=

x1 x2

¶

µ

,

A=

d2 X dt2

(6.6)

= AX where

−(k1 + k2 )/m1 k2 /m2 k2 /m1 −(k2 + k3 )/m2

¶

.

The characteristic polynomial of A is "

#

m2 (k1 + k2 ) + m1 (k2 + k3 ) (k1 + k2 )(k2 + k3 ) k22 r + r+ − . m1 m2 m1 m2 m1 m2 2

123

124

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

The discriminant of this polynomial is ∆=

½h i2 1 m (k + k ) + m (k + k ) 2 1 2 1 2 3 m21 m22 ¾

−4(k1 + k2 )(k2 + k3 )m1 m2 + 4k22 m1 m2 =

(m2 (k1 + k2 ) − m1 (k2 + k3 m21 m22

))2

(6.7)

+ 4m1 m2 k2

2

> 0.

Hence the eigenvalues of A are real, distinct and negative since the trace of A is negative while the determinant is positive. Let r1 > r2 be the eigenvalues of A and let µ

P1 =

1 s1

¶

µ

,

P2 =

1 s2

¶

be (normalized) eigenvectors with eigenvalues r1 , r2 respectively. We have m1 r1 + k1 + k2 m1 r2 + k1 + k2 s1 = , s2 = k2 k2 and, if P is the matrix with columns P1 , P2 , we have µ

P −1 AP =

r1 0 0 r2

¶

. µ

If we make a change of variables X = P Y with Y = d2 Y = dt2

µ

r1 0 0 r2

¶

y1 , we have y2

¶

so that our system in the new variables y1 , y2 is d2 y1 dt2 d2 y2 dT 2

= r1 y1 = r2 y2 .

(6.8)

Setting ri = −ωi2 with ωi > 0, this uncoupled system has the general solution y1 = A1 sin(ω1 t) + B1 cos(ω1 t),

y2 = A2 sin(ω2 t) + B2 cos(ω2 t).

Since X = P Y = y1 P1 + y2 P2 , we obtain the general solution X = (A1 sin(ω1 t) + B1 cos(ω1 t))P1 + (A2 sin(ω2 t) + B2 cos(ω2 t))P2 . The two solutions with Y (0) = Pi are of the form X = (A sin(ωi t) + B cos(ωi t))Pi =

p

A2 + B 2 sin(ωi t + θi )Pi .

125

(*) SYSTEMS OF LINEAR DIFFERENTIALEQUATIONS

These motions are simple harmonic with frequencies ωi /2π and are called the fundamental motions of the system. Since any motion of the system is the sum (superposition) of two such motions any periodic motion of the system must have a period which is an integer multiple of both the fundamental periods 2π/ω1 , 2π/ω2 . This happens if and only if ω1 /ω2 is a rational number. If X 0 (0) = 0, the fundamental motions are of the form X = Bi cos(ωi t)Pi and if X(0) = 0 they are of the form X = Ai sin(ωi t)Pi . These four motions are a basis for the solution space of the given system. The motion is completely determined once X(0) and X 0 (0) are known since µ

X(0) = P Y (0) = P

B1 B2

¶

µ

,

X 0 (0) = P Y 0 (0) = P

ω1 A1 ω2 A2

¶

.

As a particular example, consider the case where m1 = m2 = m and k1 = k2 = k3 = k. The system is symmetric and A=

k m

µ

−2 1 1 −2

¶

,

a symmetric matrix. The characteristic polynomial is r2 + 4

k2 k k k r + 3 2 = (r + )(r + 3 ). m m m m

The eigenvalues are pr1 = −k/m,pr2 = −3k/m. The fundamental frequencies are ω1 = k/m, ω2 = 3k/m. The normalized eigen-vectors are µ ¶ µ ¶ 1 1 P1 = , P2 = . 1 −1 The fundamental motions with X 0 (0) = 0 are q

X = A cos( k/m t)

µ ¶

1 1

q

,

X = A cos( 3k/m t)

µ

1 −1

¶

.

√ Since the ratio of the fundamental frequencies is 3, an irrational number, theses are the only two periodic motions of the mass-spring system where the masses are displaced and then let go. Odds and Ends

126

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

If y = f (x) is a solution of the autonomous DE y n = f (y, y 0 , . . . , y n−1 ) then so is y = f (x + a) for any real number a. If the DE is linear and homogeneous with fundamental set y1 , y2 , . . . , yn then we must have identities of the form y1 (x + a) = c2 y2 + c3 y3 + · · · + cn yn . For example, consider the DE y 00 + y = 0. Here sin(x), cos(x) is a fundamental set so we must have an identity of the form sin(x + a) = A sin(x) + B cos(x). Differentiating, we get cos(x + a) = A cos(x) − B sin(x). Setting x = 0 in these two equations we find A = cos(a), B = sin(a). We obtain in this way the addition formulas for the sine and cosine functions: sin(x + a) = sin(x) cos(a) + sin(a) cos(x), cos(x + a) = cos(x) cos(a) − sin(x) sin(a). The numerical methods for solving DE’s can be extended to systems virtually without change. In this way we can get approximate solutions for higher order DE’s. For more details consult the text (Chapter 5).

Appendix A ASSIGNMENTS AND SOLUTIONS

127

128

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

Assignment 2B: due Thursday, September 21, 2000

1 Find the solution of the initial value problem yy 0 = x(y 2 − 1)4/3 ,

y(0) = b > 0.

What is its interval of definition? (Your answer will depend on the value of b.) Sketch the graph of the solution when b = 1/2 and when b = 2 2 Find the general solution of the differential equation dy = y + e2x y 3 . dx

3 Solve the initial value problem dy x y = + , dx y x

y(1) = −4.

4 Solve the initial value problem (ex − 1)

dy + yex + 1 = 0, dx

y(1) = 1.

Solutions for Assignment 2B

1 Separating variables and integrating we get

Z

yy 0 dx x2 = + C1 4/3 2 − 1)

(y 2

from which, on making the change of variables u = y 2 , we get 1 2

Z (u − 1)−4/3 du =

x2 + C1 . 2

Integrating and simplifying, we get (u − 1)−1/3 = C − x2 /3

with C = −2C1 /3.

Hence (y 2 − 1)−1/3 = C − x2 /3. Then y(0) = b gives C = (b2 − 1)−1/3 . Since b > 0 we must have r 1 . y = 1+ (C − x2 )3

APPENDIX A: ASSIGNMENTS AND SOLUTIONS

129

If b < 1 then √ y is defined for all x while, if b > 1, the solution y is defined only for |x| < 3C. 2 This is a Bernoulli equation. To solve it, divide both sides by y 3 and make the change of variables u = 1/y 2 . This gives u0 = −2u − 2e2x after multiplication by −2. We now have a linear equation whose general solution is u = −e2x /2 + Ce−2x . This gives a 1-paramenter family of solutions ±1

y= p

Ce−2x − e2x /2

= p

±ex C − e4x /2

of the original DE. Given (x0 , y0 ) with y0 6= 0 there is a unique value of C such that the solution satisfies y(x0 ) = y0 . It is not the general solution as it omits the solution y = 0. Thus the general solution is comprised of the functions y = 0,

±ex

y= p

C − e4x /2

.

3 This is a homogeneous equation. Setting u = y/x, we get xu0 + u = 1/u + u. This gives xu0 = 1/u, a separable equation from which we get uu0 = 1/x. Integrating, we get u2 /2 = ln |x| + C1 and hence y 2 = x2 ln(x2 ) + Cx2 with C = 2C1 . For y(1) = −4 we must have C = 16 and p y = −x ln(x2 ) + 16, x > 0. 4 This is a linear equation which is also exact. The general solution is F (x, y) = C where ∂F ∂F = yex − 1, = ey − 1. ∂x ∂y Integrating the first equation partially with respect to x we get F (x, y) = yex + x + φ(y) from which and hence

∂F ∂y

= ex + φ0 (y) = ey − 1 which gives φ(y) = −y (up to a constant) F (x, y) = yex + x − y = C.

For y(1) = 1 we must have C = e and so the solution is y=

e−x , ex − 1

(x > 0).

130

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

Assignment 3B: due Thursday, September 28, 2000

1 One morning it began to snow very hard and continued to snow steadily through the day. A snowplow set out at 8:00 A.M. to clear a road, clearing 2 miles by 11:00 A.M. and an additional mile by 1:00 P.M. At what time did it start snowing. (You may assume that it was snowing at a constant rate and that the rate at which the snowplow could clear the road was inversely proportional to the depth of the snow.) 2 Find, in implicit form, the general solution of the differential equation y 3 + 4yex + (2ex + 3y 2 )y 0 = 0. Given x0 , y0 , is it always possible to find a solution such that y(x0 ) = y0 ? If so, is this solution unique? Justify your answers.

APPENDIX A: ASSIGNMENTS AND SOLUTIONS

131

Solution for Assignment 3B

1 Let x be the distance travelled by the snow plow in t hours with t = 0 at 8 AM. Then if it started snowing at t = −b we have a dx = . dt t+b The solution of this DE is x = a ln(t + b) + c. Since x(0) = 0, x(3) = 2, x(5) = 3, we have a ln b + c = 0, a ln(3 + b) = 2, a ln(5 + b) + c = 3 from which a ln

3+b = 2, b

a ln

5+b = 1. 3+b

2 2 Hence (3 + b)/b = (5 + b)2 /(3 + b) √ from which b − 2b − 27 = 0. The positive root of this equation is b = 1 + 2 7 ≈ 6.29 hours. Hence it started snowing at 1 : 42 : 36 AM.

2 The DE y 3 + 4yex + (2ex + 3y 2 )y 0 = 0 has an integrating factor µ = ex . The solution in implicit form is 2e2x y + y 3 ex = C. There is a unique solution with y(x0 ) = y0 for any x0 , y0 by the fundamental existence and uniqueness theorem since the coefficient of y 0 in the DE is never zero and hence f (x, y) =

−y 3 − 4yex 2ex + 3y 2

and its partial derivative fy are continuously differentiable on R2 . Alternately, since the partial derivative of y 3 ex + 2ye2x with respect to y is never zero, the implicit function theorem guarantees the existence of a unique function y = y(x), with y(x0 ) = y0 and defined in some neighborhood of x0 , which satisfies the given DE.

132

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

Assignment 4B: due Tuesday, October 24, 2000

1 (a) Show that the differential equation M + N y 0 = 0 has an integrating factor which is a function of z = x + y only if and only if ∂M ∂y

−

∂N ∂x

M −N is a function of z only. (b) Use this to solve the differential equation x2 + 2xy − y 2 + (y 2 + 2xy − x2 )y 0 = 0. 2 Solve the differential equations (a) xy 00 = y 0 + x, 00

(b) y(y − 1)y + y

(x > 0); 02

= 0.

3 Solve the differential equations (a) y 000 − 3y 0 + 2y = ex ; (b) y (iv) − 2y 000 + 5y 00 − 8y 0 + 4y = sin(x). 4 Show that the functions sin(x), sin(2x), sin(3x) are linearly independent. Find a homogeneous linear ODE having these functions as part of a basis for its solution space. Show that it is not possible to find such an ODE with these functions as a basis for its solution space.

APPENDIX A: ASSIGNMENTS AND SOLUTIONS

133

Solutions to Assignment 4B 1 (a) Suppose that M + N y 0 = 0 has an integrating factor u which is a function of ) ) z = x + y. Then ∂(uM = ∂(uN gives ∂y ∂x u(

∂N ∂u ∂u ∂M − )=N −N . ∂y ∂x ∂x ∂y

By the chain rule we have ∂u du ∂z du = = , ∂x dz ∂x dz so that

∂M ∂y

−

∂u du ∂z du = = , ∂y dz ∂y dz

∂N ∂x

−1 du = , M −N u dz which is a function of z. Conversely, suppose that ∂M ∂y

−

∂N ∂x

M −N

= f (z),

with z = x + y. Now define u = u(z) to be a solution of the linear DE du = −f (z)u. Then dz ∂M − ∂N −1 du ∂y ∂x = , M −N u dz ) ) = ∂(uN , i.e., that u is an integrating factor of which is equivalent to ∂(uM ∂y ∂x 0 M + N y which is a function of z = x + y only.

(b) For the DE x2 + 2xy − y 2 + (y 2 + 2xy − x2 )y 0 = 0 we have ∂M ∂y

−

∂N ∂x

M −N

=

2 2 = . x+y z

If we define

R u=e

−2dz/z

= e−2 ln z = 1/z 2 = 1/(x + y)2

then u is an integrating factor so that there is a function F (x, y) with ∂F x2 + 2xy − y 2 = uM = , ∂x (x + y)2

∂F y 2 + 2xy − x2 = uN = . ∂y (x + y)2

Integrating the first DE partially with respect to x, we get

Z F (x, y) =

(1 −

2y 2 2y 2 dx = x + + φ(y). 2 (x + y) x+y

Differentiating this with respect to y and using the second DE, we get y 2 + 2xy − x2 ∂F 2y 2 + 4xy = = + φ0 (y) (x + y)2 ∂y (x + y)2

134

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS so that φ0 (y) = −1 and hence φ(y) = −y (up to a constant). Thus F (x, y) = x +

2y 2 x2 + y 2 −y = . x+y x+y

Thus the general solution of the DE is F (x, y) = C or x + y = 0 which is the only solution that was missed by the integrating factor method. The first solution is the family of circles x2 + y 2 − Cx − Cy = 0 passing through the origin and center on the line y = x. Through any point 6= (0, 0) there passes a unique solution. 2 (a) The dependent variable y is missing from the DE xy 00 = y 0 + x. Set w = y 0 so that w0 = y 00 . The DE becomes xw0 = w + x which is a linear DE with general solution w = x ln(x) + C1 x. Thus y 0 = x ln(x) + C1 which gives y=

x2 x2 x2 x2 x2 ln(x) − + C1 + C2 = ln(x) + A +B 2 4 2 2 2

with A, B arbitrary constants. 2

(b) The independent variable x is missing DE y(y −1)+y 0 = 0. Note that y = C is a solution. We assume that y 6= C. Let w = y 0 . Then y 00 =

dw dw dy dw = =w dx dy dx dy

so that the given DE becomes y(y − 1) dw = −w after dividing by w which is dy not zero. Separating variables and integrating, we get

Z

dw =− w

Z

dy y(y − 1)

which gives ln |w| = ln |y| − ln |y − 1| + C1 . Taking exponentials, we get w=

Ay . y−1

Since w = y 0 we have a separable equation for y. Separating variables and integrating, we get y − ln |y| = Ax + B1 . Taking exponentials, we get ey /y = BeAx with A arbitrary and B 6= 0 as an implicit definition of the non-constant solutions. 3 (a) The associated homogeneous DE is (D3 − 3Dy + 2)(y) = 0. Since D3 − 3D + 2 = (D − 1)2 (D − 2) this DE has the general solution yh = (A + Bx)ex + Ce2x . Since the RHS of the original DE is killed by D − 1, a particular solution yp of it satisfies the DE (D − 1)3 (D − 2) = 0 and so must be of the form (A + Bx + Ex2 )ex + Ce2x . Since we can delete the terms which are solutions of the homogeneous DE, we can take yp = Ex2 ex . Substituting this in the original DE, we find E = 1/6 so that the general solution is y = yh + yp = (A + Bx)E x + Ce2x + x2 ex /6.

135

APPENDIX A: ASSIGNMENTS AND SOLUTIONS

(b) The associated homogeneous DE is (D4 − 2D3 + 5D2 − 8D + 4)(y) = 0. Since D4 − 2D3 + 5D2 − 8D + 4 = (D − 1)2 (D + 4) this DE has general solution yh = (A + Bx)ex + E sin(2x) + F cos(2x). A particular solution yp is a solution of the DE (D2 + 1)(D − 1)2 (D2 + 4)(y) = 0 so that there is a particular solution of the form C1 cos(x) + C2 sin(x). Substituting in the original equation, we find C1 = 1/6, C2 = 0. Hence y = yh + yp = (A + Bx)ex + E sin(2x) + F cos(2x) +

1 cos(x) 6

is the general solution. 4 (a)

Ã W (sin(x), sin(2x), sin(3x)) =

sin(x) sin(2x) sin(3x) cos(x) 2 cos(2x) 3 cos(3x) − sin(x) −4 sin(2x) −9 sin(3x)

!

so that W (π/2) = −16 6= 0. Hence sin(x), sin(2x), sin(3x) are linearly independent. (b) The DE (D2 + 1)(D2 + 4)(D2 + 9)(y) = 0 has basis sin(x), sin(2x) sin(3x) cos(x), cos(2x) cos(3x) and the given functions are part of it. (c) Since the Wronskian of the given functions is zero at x = 0 it cannot be a fundamental set for a necessarily third order linear DE.

136

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

Assignment 5B: due Thursday, Oct. 24, 2002

1 Find the general solution of the differential equation y 00 + 4y 0 + 4y = e−2x ln(x),

(x > 0).

√ √ 2 Given that y1 = cos(x)/ x, y2 = sin(x)/ x are linearly independent solutions of the differential equation x2 y 00 + xy 0 + (x2 − 1/4)y = 0,

(x > 0),

find the general solution of the equation x2 y 00 + xy 0 + (x2 − 1/4)y = x5/2 ,

(x > 0).

3 Find the general solution of the equation x2 y 00 + 3xy 0 + y = 1/x ln(x),

(x > 0).

4 Find the general solution of the equation (1 − x2 )y 00 − 2xy 0 + 2y = 0,

(−1 < x < 1)

given that y = x is a solution. 5 Find the general solution of the equation xy 00 + xy 0 + y = x,

(x > 0).

APPENDIX A: ASSIGNMENTS AND SOLUTIONS

137

Assignment 5 Solutions 1 The differential equation in operator form is (D+2)2 (y) = e−2x ln(x). Multiplying both sides by e2x and using the fact that D + 2 = e−2x De2x , we get D2 (e2x y) = ln(x). Hence 3 1 e2x y = x2 ln x − x2 + Ax + B 2 4 from which we get y = Axe2x + Be2x + 12 x2 e2x ln x − 34 x2 e−2x . Variation of parameters could also have been used but the solution would have been longer. √ √ 2 The given functions y1 = cos x/ x, y2 = sin x/ x are linearly independent and hence a fundamental set of solutions for the DE y 00 +

1 0 1 y + (1 − 2 )y = 0. x 4x

We only have to find a particular y solution of the normalized DE y 00 +

√ 1 0 1 y + (1 − 2 )y = x. x 4x

Using variation of parameters there is a solution of the form y = uy1 + vy2 with u0 y1 + v 0 y2 = 0, √ u0 y10 + v 0 y20 = x.

(A.1)

By Crammer’s Rule we have

u0 = ¯

¯ ¯ ¯

v0 = ¯

¯ ¯ ¯

¯ ¯0 ¯√ ¯ x

¯ ¯ ¯ ¯

sin √x x − sin x+2x √ sin x 2x x cos x sin √ √x x x − cos x−2x √ sin x − sin x+2x √ sin x 2x x 2x x

¯ ¯ ¯ ¯

¯ ¯ ¯ ¯

cos √ x 0 x √ − cos x−2x √ sin x x 2x x cos x sin x √ √ x x − cos x−2x √ sin x − sin x+2x √ sin x 2x x 2x x

¯ = −x sin x ¯ ¯ ¯

¯ = x cos x ¯ ¯ ¯

so that u = x cos x − sin x, v = x sin x + cos x and √ sin x cos x y = (x cos x − sin x) √ + (x sin x + cos x) √ = x. x x Hence the general solution of the DE x2 y 00 + xy 0 + (x2 − 1/4)y = x5/2 is sin x √ cos x y = A √ + B √ + x. x x 3 This is an Euler equation. So we make the change of variable x = et . The given DE becomes (D(D − 1) + 3D + 1)(y) = e−t /t

138

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

where D = DE is

d . dt

Since D(D − 1) + 3D + 1 = D2 + 2D + 1 = (D + 1)2 , the given (D + 1)2 (y) = e−t /t.

Multiplying both sides by et and using et (D + 1) = Det , we get D2 (et y) = 1/t from which et y = t ln t + At + B and y = Ate−t + Be−t + te−t ln t = A

B ln x ln x + + ln(ln(x)), x x x

the general solution of the given DE. 4 Using reduction of order, we look for a solution of the form y = xv. Then y 0 = xv 0 + v, y 00 = xv 00 + 2v 0 and (1 − x2 )(xv 00 + 2v 0 ) − 2x(xv 0 + v) + 2xv = 0 which simplifies to v 00 +

2 − 4x2 0 v =0 x − x3

which is a linear DE for v 0 . Hence

R

v0 = e Since

2−4x2 x3 −x

dx

.

2 − 4x2 −2 −1 1 = + + , x3 − x x x+1 1−x

we have

1 1 1 1 1 + − . x 2x−1 2x+1 and the general solution of the given DE is

v 0 = 1/x2 (1 − x)(x + 1) = − Hence v = − x1 +

1 2

ln

1+x 1−x

y = Ax + B(−1 +

x 1+x ln . 2 1−x

5 This DE is exact and can be written in the form d (xy 0 + (x − 1)y) = x dx so that xy 0 + (x − 1)y = x2 /2 + C. This is a linear DE. Normalizing, we get y 0 + (1 − 1/x)y = x/2 + C/x. An integrating factor for this equation is ex /x. d ex ( y) = ex /2 + Cex /x2 , dx x ex y = ex /2 + C x

Z

y = x/2 + Cxe

−x

Z

ex dx + D, x2

ex dx + Dxe−x . x2

APPENDIX A: ASSIGNMENTS AND SOLUTIONS

139

Assignment 7B: due Thursday, November 21, 2000

For each of the following differential equations show that x = 0 is a regular singular point. Also, find the indicial equation and the general solution using the Frobenius method. 1 9x2 y 00 + 9xy 0 + (9x − 1)y = 0. 2 xy 00 + (1 − x)y 0 + y = 0. 3 x(x + 1)y 00 + (x + 5)y 0 − 4y = 0.

140

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

Solutions for Assignment 7(b) 1 The differential equation in normal form is y 00 + p(x)y 0 + q(x)y = y 00 +

1 0 y + x

³

1 1 − 2 x 9x

´ y=0

so that x = 0 is a singular point. This point is a regular singular point since x2 q(x) = −

xp(x) = 1,

1 +x 9

are analytic at x = 0. The indicial equation is r(r − 1) + r − 1/9 = 0 so that r2 −1/9 = 0, i.e., r = ±1/3. Using the method of Frobenius, we look for a solution of the form y=

∞ X

an xn+r .

n=0

Substituting this into the differential equation x2 y 00 + x2 p(x)y 0 + x2 q(x)y = 0, we get (r2 − 1/9)a0 xr +

∞ X

(((n + r)2 − 1/9)an + an−1 )xn+r = 0.

n=1

In addition to r = ±1/3, we get the recursion equation an = −

an−1 9an−1 =− (n + r)2 − 1/9 (3n + 3r − 1)(3n + 3r + 1)

for n ≥ 1. If r = 1/3, we have an = −3an−1 /n(3n + 2) and an =

(−1)n 3n a0 . n!5 · 8 · · · (3n + 2)

Taking a0 = 1, we get the solution y1 = x1/3

∞ X n=0

(−1)n 3n a0 xn . n!5 · 8 · · · (3n + 2)

Similarly for r = −1/3, we get the solution y2 = x−1/3

∞ X n=0

(−1)n 3n a0 xn . n!1 · 4 · · · (3n − 2)

The general solution is y = Ay1 + By2 . 2 The differential equation in normal form is y 00 + p(x)y 0 + q(x)y = y 00 + (

1 1 − 1)y 0 + = 0 x x

so that x = 0 is a singular point. This point is a regular singular point since xp(x) = 1 − x,

x2 q(x) = x

APPENDIX A: ASSIGNMENTS AND SOLUTIONS

141

are analytic at x = 0. The indicial equation is r(r − 1) + r = 0 so that r2 = 0, i.e., r = 0. Using the method of Frobenius, we look for a solution of the form y=

∞ X

an xn+r .

n=0

Substituting this into the differential equation x2 y 00 + x2 p(x)y 0 + x2 q(x)y = 0, we get r2 a0 xr +

∞ X

((n + r)2 an − (n + r − 2)an−1 )xn+r = 0.

n=0

This yields the recursion equation an = Hence an (r) =

n+r−2 an−1 , (n + r)2

(n ≥ 1).

(r − 1)r(r + 1) · · · (r + n − 2) a0 . (r + 1)2 (r + 2)2 · · · (r + n)2

Taking r = 0, a0 = 1, we get the solution y1 = 1 − x. To get a second solution we compute a0n (0). Using logarithmic differentiation, we get a0n (r) = an (r)(

1 1 1 2 2 2 + + ··· + − − − ··· − ). r−1 r n+r−2 r+1 r+2 r+n

Hence a01 (0) = 3a0 and a0n (r) = an (r)/r + an (r)bn (r) for n ≥ 2. Setting r = 0, we get for n ≥ 2 (−1) · 1 · 2 · · · · (n − 2) a0n (0) = a0 (n!)2 from which an = −(n − 2)!a0 /(n!)2 for n ≥ 2. Taking a0 = 1, we get as second solution y2 = y1 ln(x) + 3x −

∞ X (n − 2)! n=2

(n!)2

xn = y1 ln(x) + 4x − 1 + y1 .

The general solution is then y = Ay1 + B(y1 ln(x) + 4x − 1).

142

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

Assignment 8B: due Thursday, November 23, 2000

1 (a) Compute the Laplace transforms of the functions t2 sin(t),

t2 cos(t).

(b) Find the inverse Laplace transforms of the functions s , (s2 + 1)3

1 . (s2 + 1)3

2 Using Laplace transforms, solve the initial value problem y iv − y = sin(t),

y(0) = y 0 (0) = 1, y 00 (0) = y 000 (0) = −1.

3 Using Laplace transforms, solve the system dx dt dy dt

= −2x + 3y, =x−y

(A.2)

with the initial conditions x(0) = 1, y(0) = −1. 4 Using Laplace transforms, solve the initial value problem y 00 + 3y 0 + 2y = f (t),

y(0) = y 0 (0) = 0,

where f (t) = { 1 ,

0 ≤ t < 1, − 1,

1 ≤ t < π, sin(t),

π ≤ t.

143

APPENDIX A: ASSIGNMENTS AND SOLUTIONS

Solutions for Assignment 8(b) 1 (a) d −2s 1 = 2 , ds s2 + 1 (s + 1)2 2 s −1 1 2 d s = 2 = 2 − 2 L{t cos(t)} = − ds s2 + 1 (s + 1)2 s +1 (s + 1)2 d −2s 6 8 L{t2 sin(t)} = − = 2 − 2 , ds (s2 + 1)2 (s + 1)2 (s + 1)3 8s 2s L{t2 cos(t)} = − 2 + 2 . (s + 1)3 (s + 1)2 L{t sin(t)} = −

(b)

s } = −t2 cos(t)/8 + t sin(t)/8, (s2 + 1)3 1 L−1 { 2 } = 3 sin(t)/8 − 3t cos(t)/8 − t2 sin(t)/8. (s + 1)3 L−1 {

2 If Y (s) = L{y(t)}, we have (s4 − 1)Y (s) − s3 − s2 + s + 1 =

1 . s2 +1

Hence

((s + 1)(s2 − 1) 1 + (s4 − 1)(s2 + 1) s4 − 1 ((s + 1) 1 = 4 + 2 (s − 1)(s2 + 1) s +1 1 1 1 1 3 1 1 1 s = − + − + 2 . 8s−1 8s+1 4 s2 + 1 2 (s2 + 1)2 s +1

Y (s) =

y(t) = et /8 − e−t /8 + sin(t)/2 + cos(t) + t cos(t)/4. 3 If X(s) = L{x(t)}, Y (s) = L{y(t)} we have sX(s) − 1 = −2X(s) + 3Y (s), Hence we have

sY (s) + 1 = X(s) − Y (s).

(s − 2) , (s2 + 3s − 1) (−1 − s) Y (s) = 2 (s + 3s − 1) X(s) =

and

³ X(s) =

√ 7 13 26

Y (s) = −

³ x(t) =

³√

³√

³ 1 √ s+(3+ 13)/2

+

´

13 26

√ 7 13 26

y(t) = −

´ + 12 + 12

´

1 √ s+(3+ 13)/2 √ −(3+ 13)t/2

+ 12 e

13 26

´

+

√ −(3+ 13)t/2

+ 12 e

³√

³ + +

´

√ −7 13 26 13 26

+

13 26

√ 1 , s−( 13−3)/2

´

√ −7 13 26

³√

1 2

+ 1 2

+

´ 1 2

e

´

+

1 2

√ 1 , 13−3)/2 √ ( 13−3)t/2

s−(

e

,

√ ( 13−3)t/2

4 We have y 00 + 3y 0 + 2y = 1 − 2u1 (t) + (sin(t) + 1)uπ (t). If Y (s) = L{y(t)} (s2 + 3s + 2)Y (s) =

e−s 1 −2 + e−πs 2 s

³

´

−1 1 + , +1 s

s2

144

FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS

since sin(t + π) = − sin(t). Hence Y (s) =

Since

1 −2 + e−s s(s + 1)(s + 2) s(s + 1)(s + 2) µ ¶ −1 1 −πs +e + . (s2 + 1)(s + 1)(s + 2) s(s + 1)(s + 2)

1 1 1 1 = − + , s(s + 1)(s + 2) 2s s+1 2(s + 2) 1 1 1 − 3s 1 = − + , (s2 + 1)(s + 1)(s + 2) 2(s + 1) 5(s + 2) 10(s2 + 1)

we have

³

´

1 1 −1 2 1 1 − + + + − e−s 2s s + 1 2(s + 2) s s + 1 s + 2 ¶ µ 1 3 7 1 3s + − + − − e−πs . 2s 2(s + 1) 10(s + 2) 10(s2 + 1) 10(s2 + 1) Y (s) =

and

³

´

1 e−2t − e−t + + − 1 + 2e1−t − e2−2t u1 (t) 2 µ 2 π−t ¶ sin(t) 3 cos(t) 1 3e 7e2π−2t + − + + − uπ (t). 2 2 10 10 10 y(t) =

Hence

y(t) =

1 1 − e−t + e−2t , 2 2 − 1 + (2e − 1)e−t + ( 1 − e2 )e−2t , 2

2

1 7 2π −2t 3 e )e (2e − 1 − eπ )e−t + ( − e2 + 2 2 10 1 3 +

10

sin(t) −

10

cos(t),

0 ≤ t < 1, 1 ≤ t < π,

π ≤ t.

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