ORDINARY - Islamic University of Gaza

ORDINARY DIFFERENTIAL EQUATIONS AND STABILITY THEORY

AN INTRODlJCTION

by David A. Sanchez

University of New Mexico

Dover Publications, Inc. New York

Copyright © 1968 by W. H. Freeman and Company. All rights reserved under Pan American and Inter·

national Copyright Conventions.

Published in Canada by General Publishing Company, Ltd., 30 Lesmill Road, Don Mills, Toronto, Ontario.

Published in the United Kingdom by Constable and Company, Ltd., 10 Orange Street, London WC2H 7EG.

This Dover edition, first published in 1979, is an unabridged and unaltered republication of the work originally published in 1968 by W. H. Freeman and Com· pany. The Dover edition is published by special arrangement with W. H. Freeman and Company, 660 Market St., San Francisco, Calif. 94104.

International Standard Book Number: 0-486-63828-6 Library of Congress Catalog Card Number: 79-52007

Manufactured in the United States of America Dover Publications, Inc.

180 Varick Street New York, N.Y. 10014

Preface

In this book I have attempted to give a brief, modern introduction to the subject of ordinary differential equations, with an emphasis on stability theory. This emphasis has been directed chiefly toward the undergraduate or beginning graduate student, who is often deprived of any exposure to the newer concepts .

Neglect of these concepts is usually justified by the argument that the student is insufficiently equipped mathematically to handle them. As a result, the usual course in ordinary differential equations consists in learning a battery of special techniques to solve special equations ; possibly an existence theorem is proved .

This is an injustice to the student, for in actuality a modicum of knowledge beyond the calculus will carry him a relatively long way in the study of stabil ity theory . If his appetite i s whetted, it should serve as an i ncentive to equip h imself to proceed further. In any case the student wi l l be aware that the finding of solutions of exotic ordinary differential equations is not the principal aim of the subject.

Chapters I, 2, 3, and 6 consider the usual problems of existence and uniqueness of solutions, the maximum interval of existence, fundamental systems of solutions of l inear equations, and nonhomogeneous l inear equations. I chose to begin with a discussion of the first-order l inear system for two reasons. First of all , the results for the nth-order l inear equation fol low with almost no effort. Second, and most important, the notion of a fundamental matrix is developed.

VI Preface

This is used in the representation of solutions of homogeneous and

inhomogeneous systems, and plays a key role in the discussion of

stability of nonautonomous systems.

Chapters 4 and 5 are introductory discussions of stability theory

for autonomous and non autonomous systems. I have made no at

tempt to be encyclopedic, but merely give some basic results. These

are a matter of personal choice and of the intended audience, but

hopefully they will serve the ambitious reader as a suitable starting

point in this vast field.

The problems given are, for the most part, designed to fill out

the text material. Appendix A deals with series solutions near regular

singular points. The proof of the main theorem requires some know

ledge of complex variables; it may be omitted and the reader may

proceed directly to the examples given. Appendix B gives some results

dealing with periodic solutions. The books listed as references were

extremely useful to the author, and the reader will be well rewarded

in investigating their contents.

I wish to thank Professors K. S. Williams and J. D. Schuur,

Doctor J. L. Brenner, and Miss Patrice Whittlesey for their comments

and suggestions. My appreciation is extended to the Mathematics

Departments of the University of Manchester and of the University

of California at Los Angeles for technical assistance, and to Mrs.

Audrey Biggar and Mrs. Ruth Goldstein for typing. Finally I wish

to thank the undergraduates who endured my preliminary classroom

efforts to give ordinary differential equations a modern meaning.

Los Angeles, California May 1967

D. A. SANCHEZ

Contents

CHAPTER 1 Introduction

1 . 1 Preliminary Notation

1 .2 The Ordinary Differential Equation

1 .3 An Existence and Uniqueness Theorem

1 .4 The Maximum Interval of Existence

Problems

CHAPTER 2 The Linear Equation: General Discussion

1 4 8

10 13

2. 1 Introduction 15

2.2 Fundamental Solutions 17 2.3 The Wronskian 21

2.4 The Nonhomogeneous Linear Equation 25

2.5 The nth-Order Linear Equation 28

2.6 The Nonhomogeneous nth-Order Linear Equation 33 Problems 37

CHAPTER 3 The Linear Equation with Constant Coefficients


3.2 The Nonhomogeneous nth-Order Linear Equation 51 3.3 The Behavior of Solutions 55 3.4 The First-Order Linear System 58

Problems 65

CHAPTER 4 Autonomous Systems and Phase Space

4.1 Introduction 68

4.2 Linear Systems-Constant Coefficients 73

403 A General Discussion 81

4.4 Nonlinear Systems 84

Problems 88

CHAPTER 5 Stability for Nonautonomous Equations

5.1 Introduction 91

5.2 Stability for Linear Systems 93

5.3 Two Results for Nonlinear Systems 99

5.4 Liapunov's Direct Method 105

5.5 Some Results for the Second-Order Linear Equation 111

Problems 117

CHAPTER 6 Existence, Uniqueness, and Related Topics

6.1 Proof of the Existence and Uniqueness of Solutions 121

6.2 Continuation of Solutions and

the Maximum Interval of Existence 130

6.3 The Dependence of Solutions on

Parameters and Approximate Solutions 1 36

Problems 1 41

APPENDIX A Series Solutions of Second-Order Linear Equations 144

APPENDIX B Linear Systems with Periodic Coefficients 155

References 161

Index 1 63

ORDINARY DIFFERENTIAL EQUATIONS

AND STABILITY THEORY AN INTRODUCTION

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CHAPTER

1 Introduction

1.1 Preliminary Notation

For convenience, we will employ vector notation throughout the text, and n-dimensional vector-valued functions of n- or (n + 1)dimensional vectors will be most frequently used . Thus if the positive integer n is unspecified, then any results stated will be applicable to one- or many-dimensional problems.

In general, suppose we are given the functionJmapping a subset of Rm, Euclidean m-dimensional space, into Rn, Euclidean n-dimensional space . If x = (x I, ... , xm) i s in the domain of J, and we denote its image under J by Y = (Y I ' ... , Yn), then we may write

Y = J(x) = (f,(x), '" ,fn(x»,

where we define i= I, ... ,n.

We will say that J is continuous in x if each Ji is continuous in x. Furthermore, we define the vector of partial derivatives as

:� = G�> ... , ��:), where I ::; j ::; m.

Given the n-dimensional vector x(t) = (xt(t), ... , xit», where t is a real variable and each xi(t) is real-valued, we say x(t) is continuous at t = to if each xi(t) is continuous at t = to, and it i s

2 Introduction

differentiable if each xj(t) is differentiable. We then may express the derivative vector as

dx x(t) = - = (XI(t), ... , xn(t),

dt

and successive derivatives will be denoted by x(t), X(3)(t), ... , X(k)(t). If x(t) is given as above, we denote the norm of x(t) by

n I lx(t) 1 1 = L Ixi(t)l, i= I

and for each t this is a mapping of x(t) into the nonnegative real numbers . It has the properties (i) Ilx(t)11 = 0 if and only if x(t) = 0, that is, each x/t) i s zero; (ii) Ilkx(t)II = Ikll l x(t)1 1 for any real or complex scalar k; and (iii) I l x(t) + y(t)11 :::; I l x(t) 11 + I ly(t) l l . The above norm has certain computational advantages over the usual Euclidean norm, { n } 1/2

I lx(t)11 = J! Ixi(tW , which also satisfies the characteristic properties of the norm l isted above. For geometrical convenience (for example, when using polar coordinates) we will occasionaIIy use the latter norm; any results given wiII not, however, depend on the norm chosen.

Frequently in the text we wiII be considering a given function J mapping a subset of Rn+1 i nto Rn. If we denote a point i n Rn+1 by (t, x), where t is real and x = (XI' ... , xn), then its image, wherever defined, may be denoted by

y = (YI ' . . . , Yn) = J(/, x) = (fl(t, x), ... ,//1, x».

In particular, if x = x(t) = (xl(t), ... , xn(t», then y = yet) = J(/, x(t» is an element of W dependent on the real variable t.

If J(t, x(t» is continuous for (say) II :::; t :::; 12, then we can define the integral

{2J(t, x(t» dt = ({2JI(t, x(t» dt, ... , (2JnCt, xCt» dt) ,

and the usual rules of integration wiII hold.

1.1 Prel iminary Notation

Example: S ince

I I,

I I, f fief, x(t» dt :-:; f If/f, x(t)I dt for i = 1, ... , n,

tl tl

then

wh ich is equivalent to the statement

IIf2 J(t, x(t» dt 11:-:; f' IIJ(t, x(t» II dt.

3

We will also consider complex-valued functions z(t) = u(t) + iv(t), where u and v are real-valued functions of the real variable t. Then we say that z(t) is continuous at t = fo if u(t) and vet) are continuous at f = to, and z(t) i s differentiable if u(t) and vet) are differentiable. The derivative of z(t) i s given by z(t) = li(t) + iv(t), and the usual roles of differentiation apply. In addition we define the usual complex modulus :

Iz(t)1 = [u2(t) + v2(t)P/2.

By a scalar function we will always mean a real- or complex-valued function of the real variable t.

Example: Let x = (XI' Xl) belong to R2, and then define the function f mapping R3 i nto Rl by y = f(t, x) = (fXIXl, 3tlxI + Xl)'

Hence YI = fl(t, x) = tXlx2, Y2 = fl(t, x) = 3t2xI + Xl' If xl(t) = t and xz(t) = cos t, then x(t) = (t, cos t) and y(t ) = f(/, x(t» = (tl cos I, 3t3 + cos t). Then, for i nstance, yet) = (2t cos f - tl s in I, 9tl - sin t)

and 1</1 (n2 3n4 ) f y(t)dt = - - 1, - + 1 . o 4 64

4

If, instead, xt(t) = t and X2(t) = t + i sin t, then x(t) = (t, t + i sin t)

and yet) = /(t, x(t»

= [t3 + il2 sin t, (3t3 + t) + i sin tl.

In this case, for instance, yet) = [3t2 + i(2t sin t + t2 cos t), (9t2 + I) + i cos tl.

Introduction

Finally, we wiII not distinguish between the scalar ° and the zero vector having zero as each of its components. The meaning will always be clear from the context.

1.2 The Ordinary Differential Equation

The subject of study is an ordinary differential equation-an equation containing the derivatives of an unknown function x(t), with t a real variable, and possibly containing the unknown function itself, the independent variable t, and given functions. In addition, initial conditions, which the unknown function is required to satisfy, may be given. With such an equation, the object is two-fold: (i) to find the unknown function or class of functions satisfying the equation, and (ii) whether (i) is possible or not, to gain some information about the behavior of any function satisfying the equation.

DEFIN ITION: The order of a differential equation is the order of the highest derivative of the unknown function appearing in it.

Therefore the general form of an ordinary differential equation of kth order is

F(t, x, x, . . . , X(k» = 0, (1) where x = x(t) = (xt(t) , ... , xit» is an unknown function, and F is a function defined on some subset of R"(k+ 1) + 1.

1.2 The Ordinary Differential Equation 5

DEFI N I T I O N : A function x = <p(t) = (<PI(t), ... , <Pn(t», rl < t < r2, which when substituted in ( 1) reduces it to an identity, is called a solution of ( 1 ) , and (rl ' r 2) is its interval of definition. Since we assume that k � 1 , it follows that <pet) is differentiable

and hence continuous on rl < t < r2• If the domain of the function F is some set B in Rn(k+ 1)+ 1 , then the point (t, <p(t), cp(t), ... , <p(k)(t» belongs to B for rl < t < r2•

Very little can be said about the equation in the form given in (I), so let us assume that we can solve (locally) for X(k). We obtain

(k)-G(t ' (k-l» (2) x - , x, x, . . . , x ,

the kth-order equation in normal form. In this case, since X(k) is an n-dimensional vector, the function G is a mapping from some subset of Rkn+ 1 into Rn.

Examples

(a) x = a(t) x + b(t), where aCt) and bet) are given scalar functions and x = x(t) i s an unknown scalar function, is a first-order equation, and G(t, x) = a(t)x + bet) is a mapping from R2

into RI. (b) X(4) = [a(t)x(3) + b(t)X

2]1/2, where aCt) and bet) are given scalar

functions and x = x(t) is an unknown scalar function, is a fourthorder equation, and G(t, x, x, x, X(3» = [a(t)x(3) + b(t)X

2]1/2

is a mapping from R5 into Rl. ( c) The system

Xl = tX I +2X2 +x3,

x2 = t2x2 + (X3)2,

X3 = 5xI + 2X2 + t3x3 + et, where x = x(t) = (xl (t), x2(t), x3(t» i s an unknown vector function in R3, is a first-order equation , and G(t, x) = (Gt(t , x), G2(t, x), G3(t, x»

= (tXt + 2X 2 + x3, t2

X2 + (X3)2

, 5xI + 2X 2 + t3X3 + et) is a mapping from R4 into R3 .

6 Introduction

It should be noted that an equation of the form (2) with n > 1 is sometimes called a kth-order system.

Given an equation in the form (2) with k > 1 , the following substitution reduces it to an equation with k = 1. Let YI = x and Y2 = X, ... , Yk = X(k-l); then J\ = x = Y2' Y2 = X = Y3' ... , Yk-l =

(k-l) _ d ' - (k) - G( . (k-l» - G( x - Yk' an Yk - X - t, x, x, . . . , x - t, Y!, Y2 , . . . , Yk) = G(t, y). This is the first-order system Y = J(t, Y), where Y = (Yl' ... , Yk) and J(t, y) = ( fl(t, Y), . . . , f,..(t, y» = (Y2, . . . , Y k' G(t , y). Note that if x is an n-dimensional vector, then Y is a (k x n)

dimensional vector.

Examples

(a) Given any second-order equation x = get, x, x), where x = x(t) i s an unknown scalar function, and letting YI = x, Y2 = x, we have the system Y! = Y2' Y2 = g(t, YI' Y2). If we let Y2 = y, this can be written, x = y, Y = get, x, Y), a first-order equation, where (x, y) = (x(t), yet»� is an unknown two-dimensional vector. In the first equation a solution would be a scalar function ; in the second equation it is a pair consisting of a scalar function and its first derivative.

(b) The second-order system XI = aXI + bx2, X2 = eX! + dX2' wherex(t) = (xl (t), X2(t» i s an unknown two-dimensional vector, is transformed as follows. Let (YI' Y2) = x = (Xl' X2) and (Y3, Y4) = x = (X I ' x2) = (YI ' Y2)' Then we have the first-order system

where Y = y(t ) = (Yl(t), Y2(t), Y3(t), Y4(t » i s an unknown four

dimensional vector function.

It follows that we need only consider the first-order equation

x = J(t , x), (3)

where x = x(t) is an unknown n-dimensional vector function, and J(t, x) is a mapping from a subset of Rn+1 into Rn.

Given the first-order equation (3), let us assumed that J(t, x) is defined in a domain B, an open connected set in Rn+ I. If J(t, x) i s

1.2 The Ordinary Differential Equation 7

continuous in B, and x = cp(t) = (CPI(t), ... , cpit», r1 < t < r2, is a solution of (3), then it may be thought of as a curve lying entirely i n B. Furthermore, it will have a continuously turning tangent at each point given by

cp(t) = (CPI(t), ... , cpit» = (fl(t, cpU», ... ,f;,(t, cp(t»).

Such a curve is often called an integral curve. Given a point (t, x) i n B, we may compute the value of the

vector I(t, x). If we construct a line segment CPt,x passing through (t, x) and paraIlel to l(t, x), and do this for all (t, x) in B, we obtain the direction field of (3). It fol lows that any integral curve cp(t),

rl < t < r2, of (3) is then tangent to CPt,<p(t) at each point (t, cp(t» of B.

x x B

@ \ /:::/ --Integral curves Direction field

Of particular i nterest wil l be the first-order system

n Xi = I aij(t)xj + hi(t),

j=1 i = 1, ... , n,

where aij(t), i, j = I, . .. ,n, and b;(t), i= I, . .. , n, are continuous real-va lued functions on rl < t < r2 and x(t) = (xl(t), ... , xn(t» is an unknown n-d imensional vector. If we denote by A(t) the n x n

matrix (aJt)), and by B(t) the vector (bl(t), ... , bn(t)), then the system can be conveniently expressed as

x = A(t)x + B(t).

Hence l(t, x) = A(t)x + B(t) and is defined in the infinite slab

B = ret, x) I rl < t < r2, - 00 < Xi < 00 , i = 1, ... , n}.

8 Introduction

Observe that the form of equation (3) also suffices for the case of complex-valued unknown functions. For given the equation i = f(t, z) , where z = x + iy, it can be written x + iy = f(t, z) = h(t, x, y) + ig(t, x, y). This is equivalent to the system of equations x = h(t, x, y), y = g(t, x, y), where x = x(t) and y = y(t) are unknown functions, and i s therefore in the form (3).

1.3 An Existence and Uniqueness Theorem

We now state a theorem giving sufficient conditions for the existence and uniqueness of solution of the first-order equation i n normal form. The proof of the theorem is deferred until the last chapter inasmuch as the method of proof leads to several other results not relevant to our present discussion.

T H E O R E M 1.3.1. Let the equation (*) x = f(t, x) be given, where f(t, x) is defined in some domain B contained in Rn+ 1. Suppose in addition that f and of/oxi' i = I, . .. , n, are defined and continuous in B. Then for every point (to , xo) in B there ex ists a unique solution x = q>(t) of (*) satisfy ing q>(to) = X o and defined in some neighbor hood of (to, xo)·

Some remarks are in order. First of all, by unique is meant the following.

If two solutions x = q>(t) and x = lj!(t) of the equation (*) both satisfy q>(to) = lj!(to) = xo , then these solutions are identical in their common interval of definition.

Hence the theorem states that through every point of B there passes one and only one integral curve.

D EFI N I T I O N : The pair (to, xo) are called the in itial values of the solution x = q>(t). The relation X o = q>(to) is called the initial condition for the solution q>(t).

The existence and uniqueness theorem is useful in the following sense: suppose by some technique we are able to find a family K of solutions of x = f (t, x). Furthermore, given a point (to, xo) in B, there is an element q> in K satisfying q>(to) = Xo. If the hypotheses of

1.3 An Existence and Uniqueness Theorem 9

Theorem 1.3.1 are satisfied, then by uniqueness K must describe all solutions, and we need look no further for other solutions.

Example : Consider the equation x = J(t),

where x = x(t) i s an unknown scalar function, and J(t) is continuous on a < t < b . The hypotheses of Theorem 1.3.1 are satisfied and B is the infinite strip B = {(t, x) I a < t < b,

- 00 < x < oo}. Consider the family K described by

o/(t ; to , xo) = Xo + r J( s)d s, to

a < to , 1 < b, - 00 < Xo < 00.

Any element of K satisfies 1>(t ; to , xo) = J(t ) and hence is a solution ; given any (to , xo) i n B, then the element 0/(/; to , xo ) satisfies o/(to ; to , xo) = Xo . Therefore K describes all solutions of the equation.

Given any to in (a, b), we let F(t) = r J( s) d s, a < t < b, to

then any solution 0/(1) satisfying the initial conditions 0/(11) = Xl is given by o/(t) = XI - F(t l ) + F(t), so all solutions are obtained by a translation of F(t ) .

The integral curves look like this .

x

o

I� IB I I I I I (j. I I f\.tY � I I f\.t) I

Integral curves of x = f(t)

In the above example we were able to find an explicit form of the solution. The class of d ifferential equations for which this can be done is small, but if we have a differential equation for which we know a solution exists, we may proceed to investigate properties of

10 Introduction

the solution (such as behavior for large t or bounded ness), regardless of whether we know its explicit form. The qualitative study of solutions of differential equations thus depends upon existence.

Uniqueness would be of importance if, for instance, we wished to approximate the solution numerically. If two solutions passed through a point, then successive approximations could very well jump from one solution to the other-with misleading consequences. Furthermore, uniqueness assures us that the domain B i s smoothly covered by a nonintersecting family of integral curves. Such a fact i s useful in the study of dynamic or spatial properties of solutions.

It should be remarked that if in the hypotheses of Theorem 1.3.1 we only require that fbe continuous in B, then existence of a solution satisfying (p(to) = Xo i s still guaranteed. However, uniqueness may no longer occur.

Example: The functions ({J(t) = (t - to)3 and tfJ(t) = 0, - w < to, t < W, are easily seen to be solutions of the equation x = 3X2/3. Hence, given the initial values (to, 0), - w < to < w, there are two solutions satisfying them. Note that f(x) = 3X

2/3

is continuous at x = 0, but df/dx = 2X-I/3 fails to exist there . For the domains B = {(t, x)l- w < t < w, x > O} or B = {(t, x)1 - w < t < w, x < O}, both f and dfldx are continuous, and the unique solution satisfying ({J(to) = Xo for Xo "# 0 is given by ({J(t) = (t - to + X�/3 )3. It is defined for - w < t < w, but fails to be unique at t = to - x�/3, where it intersects the zero solution.

1 .4 The Maximum Interval of Existence

Suppose we are given two solutions ({Jt(t), r1 < t < r2, and ((J2(t), SI < t < S2 , of the differential equation x = f(t, x), and both solutions satisfy the initial condition. Therefore ({Jt(to) = ({J2(tO) = Xo. If the equation satisfies the hypotheses of Theorem 1.3.1, then in a neighborhood of (to , xo) we have uniqueness, and the two solutions overlap. That is , if for instance r1 < Sl < 10 < r2 < S2' then (f>l(t) = ({J2(t) for Sl < 1 < r2·

1.4 The Maximum Interval of Existence 1 1

However, we can define a new solution <p( t) defined on r l < t <S2 , and containing both If>1( t) and If>2( t), as follows :

<p( t) = <PI( t)

If>( t) = If>2( t)

i f

if

It i s a solution, s ince <PI ( t) and <P2( t) are solutions, it agrees with their common values on Sl < t < r2 , and it is defined on the larger interval r l < t < S2. This same procedure of tacking together solutions would apply if we were given a finite number of solutions If>1(t) , . . . , <Pm( t), such that <P I (to) = . . . = <Pm(tO) = xo · We could then define a new solution <p( t) satisfying <p(to) = xo and whose interval of definition contains those of <PI( t), . . . , <Pm( t) .

We might conjecture that, given in itial values ( to , xo), there exists a solution <p( t), ml < t < m2 , whose interval of definition (ml, m2) is maximal in some sense . The following theorem indicates that this is true.

THE ORE M 1.4.1. Suppose the hypo theses o f Theorem 1.3. 1 ar e satisfied for the d ifferential equa tion (*) x = f( t, x). Then , give n initial l'a lues ( to , xo), there ex ists a solution <p( t) o f (*) , de fined on m I < t < m2 , satis fying .<P(to) = Xo ; fur thermore , if I/I( t) is any other solution and I/I( to) = Xo . then it s interval o f definit ion is conta ll1ed in (m I' m2).

Proo f : Let M be the set of all intervals of definition of solutions of (*) satisfying the initial values, and M is not empty, since at least one such solution exists. Let MI be the set of all left end points of the elements of M and M2 the set of all right end points. Let ml = i nf MI and m 2 = sup M2 and suppose tl belongs to (m l, m2) .

Then there exists a solution I/I(t) whose interval of definition contains 11 and we define <P(tl) = 1/1(11) . By uniqueness, <p( t) is wel l-defined on m l < t < m 2 and i s a solution, since for every t it agrees with a solution, and finally <p(to) = xo . By the construction, (m l , m2) i s maximal i n the sense described .

DEFI N ITIO N : The interval (m l, m2) i s called the maximum interval of existence corresponding to the initial values ( to , xo) .

12 Introduction

Examples

(a) If we consider the equation x = J(t) previously discussed in Section 1.3, and J{t) i s continuous on a < t < b, then, gIven initial values (to, x o), the corresponding solution is

I <pet) = Xo + f J(s) ds.

10

The maximum interval of existence is (a, b), since the integral i s by hypothesis only defined for a < t < b. For instance, i f J(t) = e r, then (a, b) = ( - 00, 00), whereas if

1 J(t) = t(l _ t) ,

then (a, b) can be (0, 1), (1, 00) or ( - 00 , 0), depending on the choice of to .

(b) This example shows that the maximum interval of existence may vary considerably with the choice of initial values. Given the equation x = - 3X4/3 sin t, xU) an unknown scalar function, i ts solutions are x(t) = 0 and x(t) = (c - cos t)-3, where the constant e is determined by the choice of initial values (t o, x o)' The hypotheses of Theorem 1.3.1 are satisfied where B is the whole (t, x) plane.

The solution x{t) = 0 is defined on - 00 < t < 00, and hence any initial value (to, 0) has ( - 00, 00) as its maximum i nterval of existence. The second solution is defined on - 00 < t < 00 if lei > 1, whereas it will only be defined on a fin ite interval if lei:::; 1 . For example, the solution x{t) satisfying x(nI2) = 1 /8 i s

x(t) = (2 - cos t)-3

, - 00 < t < 00, whereas the solution satisfying x{nI2) = 8 is

x(t) = (1/2 - cos t)-3, nl3 < t < 5n13, and both the intervals are maximal for their respective in itial values.

A sketch of the solutions with initial values (n/2, 0), (n/2, 1/8) and (n/2, 8) would look like the figure on page 13.

Problems 13 x

x(t) = H - cos W 3

8

x(t) = (2 - cos tj-3

x(t) = 0

Solutions of x = - 3x'" sin t

Problems

t. Write the following equations as a first-order system in normal form. (a) X(4) + X(3) cos t - x + x2 sin t = 0, x = x(t) a scalar function.

(b) ii + u = v sin t, v + v = it cos t, u = u(t), v = vet) scalar functions.

2. G iven the equation x = I(x), where x = x(t) is an unknown scalar

function, and I and dl/dx are defined and continuous on the strip B = {(t, x) I - 00 < t < 00, a < x < b}. Assume I(x) 1= 0, a < x < b,

and let

" 1 F(x) = tf(s) ds,

where a < Xo < b. Show all solutions are described by the family

x(t) = cp(t - c), where cp is the inverse of F, and the constant c is determined by the initial conditions.

3. Given the equation x = I(t)g(x) == F(t, x), where x = x(t) is an

unknown scalar function, and F and aF / ax are defined and con

tinuous on the rectangle B = {(t, x) I t1 < t < fz , a < x < b}. Assume

that g(x) 1= 0, a < x < b.

Let u = cp(t) be the solution of it = l{t) and x = ifJ(u) be the

solution of dx/du = g(x). Show that all solutions of x = F(t, x) are

14 Introduction

described by the family x(t) = !fi(q;(t) - c) , where the constant c is

determined by the initial conditions.

4. If I(t, x) is homogenous of degree zero (that is, /(!XI, !Xx) = 1(/, x) for

!X i:- 0), show that the substitution x = lu or x = t/u, X(/), u(t) scalar

functions, transforms the equation x = 1(/, x) into the type con

sidered in Problem 3.

5. Find al l solutions of the following differential equations, where

x = x(t) is a scalar function. Discuss possible choices for the domain

B of Theorem 1.3. 1 .

(a) x = Ie'. (e) x = -(I + I)x/t.

(b) x = I log(t2 - 1 ). (f) x=t3(x+ 1)-2.

(c) x =x2 -4. (g) x = (x + 1)/1.

(d) x = sec x. (h) x = (x -V x2 + t2)/t.

6. Describe all solutions of x = (x2 - I)/xl, x = X(/) a scalar function,

and find the maximum interval of existence of the solution satisfying

x(1) = V2/2. 7. Solutions need not be given as explicit functions of t. For example,

show that a solution of x = (I + I)(x + I)/IX, x = X(/) a scalar func

tion, is given by I(X + I) = 2�-'.

8. Given the Riccati equation x = p(/)X + q(/)x2 + r(/), where x = X(/) is a scalar function and p, q, and r are continuous on a < I < b. Show

that if x = q;(t) is a solution, then there are further solutions of the

form x = q;(t) + IN(t), where !fi(t) is a solution of an equation of the

form x = c(t)x + d(t).

C H AP T ER

2 The Linear Equation:

General Discussion

2.1 Introduction

In this chapter we will discuss the n-dimensional first-order system of equations

n Xi = I Qi/t)Xj + Mt), j=1 i = 1, . . . , n, (1)

where x(t) = (xl (t), ... , xn(t» is an unknown vector function , and 0ij(t) and bi(t), i,j = 1, ... , n, are given continuous functions on 'I < t < '2·

As remarked in Section 1 .2, this equation can be conveniently written as

X = A(t)x + B(t), (2) where A(t ) is the n x n matrix (oij(t» and B(t ) is the vector (bl (t), . . . , bn(t». The function f (t, x) = A(t)x + B(t) satisfies the hypotheses of Theorem 1.3.1 in the domain

B = {(t, x) 1'1 < t < '2' - 00 < Xi < 00, i = 1, . . . , n}.

Therefore, given i nitial values in B, there exists a unique solution of (2) satisfying them.

16 The Linear Equation: General Discussion

Examples

(a) A number of physical processes involving growth or decay are described by the differential equation

x = a(t)x.

Here x = x(t ) is an unknown scalar function representing, for example, the population or amount of material remaining at time t , and a(t ) i s a continuous function on 0 < t < 00 and represents a growth or decay factor.

The equation i s of the form (2), with A(t ) = aCt ) and B(t ) = 0, and the solution (see Problem 3, Chapter 1) satisfying x(t o) = Xo i s given by

x(t) = Xo exp [( a(s) dS]. (b) The nth-order equation

in) + a 1(t)y(n-l) + ... + an(t)y = bet), where y = yet ) is an unknown scalar function and ai(t ),

i = 1 , . . . , n, and b(t ) are continuous functions on rl < t < r2, is of extreme importance both mathematically and in physical applications.

By the substitution (see Section 1 .2) x I = y, x2 = y, ... , Xn = in-I), the equation can be written in the form (2), where

0 1 0 0 0

0 0 1 0

A(t) = B(t) =

0 0 1 0

- an(t) -a l(t) bet)

The following theorem gives an important property of solutions of (2). Its proof is deferred until the last chapter, since it i s a consequence of the proof of the existence and uniqueness theorem.

T H E O R E M 2 . 1 . 1 . Given the equat ion x = A(t ) + B(t ), l-rhere A(t )

and B(t ) are cont inuous on r1 < t < r2, then for any initial

2.2 Fundamental Solutions 17

value (to , x o), rl < to < r2, there ex ists a solut ion defined on rl < t < r2 and sat isfying the in itial value.

The theorem states that the maximal interval of continuity of A(t) and B(t) is the maximum interval of existence for any initial value. Thus, for instance, if ai(t), i = I, ... , n, and b( t) are continuous for al l t, then any solution y(t ) of the equation given in Example (b ) can be defined for - OCJ < t < OCJ •

2.2 Fundamental Solutions

We will first discuss the first-order l inear homogeneous system given by

x = A(t)x, (3)

where x = x (t ) = (xl(t ), ... , x it» i s an unknown n-dimensional vector function, and A(t) = (aiit » is an n x n matrix that is continuous on rl < t < r2' From the previous discussion it fol lows that given (to , x o), rl < to < r2, there exists a unique solution x(t), defined on rl < t < r2 and satisfying x(to) = Xo .

The system (3) is of the form discussed in the previous section and is called homogeneous, since B(t) = O. It i s called l inear, since any linear combination of solutions of (3) is also a solution. Specifically, let <Pi(t) = (<PI i(t), ... , <Pni(t», i = I, . . . , m, be solutions of (3), and let ci, i = I, . . . , m, be arbitrary constants. Then, for <p(t ) =

I7'=1 Ci<P/t), we have

by the properties of matrix-vector multipl ication. But the relation cp = A(t)cp implies that cp(t ) i s also a solution of (3).

Note that <p(t) == 0, rl < t < r2, is a solution of (3), and in fact is the only solution satisfying <p(to) = 0 for rl < to < r2, as the following lemma shows.

L E M M A 2.2. 1 . If rl < to <'2 and x = <p(t) is a solut ion of (3) sat isfying <p (to) = 0, t hen <p(t) is identi cally zero on r1 < t < '2'

Proof The solution x( t ) == 0, r 1 < t < r 2' satisfies x( to) = 0, but by uniqueness we must have X( /) = <p(t), s ince all solutions are defined on rl < I < r2 .

We now introduce the key notion of l inear independence of a given collection of scalar or vector-valued functions.

D EF I N I T I O N: A collection of functions IXI(/), ... , IXm(t), a < I < b, is linearly independent if there exist no constants CI , • • • , Cm , not all zero, such that LJ= I CjlXj(t) i s identically zero on (a, b) . The collection is linearly dependent if it is not l inearly independent.

Examples

(a) The collection of scalar functions I, t, t2, ... , tm, - 00 < t < 00 , is

l inearly independent, since P m(t) = LJ= 0 C j fj

== 0, - 00 < f < 00 , can only occur when Co, CI, ... , Cm are all zero, since a nontrivial polynomial can only have a finite number of zeros.

(b) The collection of n-dimensional vector functions jth place

lXi t)=(O, ... ,O, 1, O, .. . ,O)=ej,

j = I, . . . , n, - 00 < t < 00, is linearly independent, since LJ=ICjlX/t)=(CI, ... ,cn) = O if and only if cj=O for j = 1, ... , n.

(c) The vector functions IX I (t) = (Ji f, cos f), ct2(t) = (J2 t2, sin I) and ct3(t) = (I - f2, cos (t + n/4» are l inearly dependent on - 00 < t < 00, since L7=1 cjlXit) == 0, - 00 < t < 00 , for

CI = J2/2, C2 = - J2/2, and C3 = - I.

LE M MA 2.2.2. If <PI(t), ... , ipm(t), rl < 1 < r2, are a collection of linearly independent solutions of (3), then the linear combination LJ= I Cjip/t) naer vanishes on rl < f < r2 unless CI = .. . =

Cm =0.

Proof. If we let ip(t) = LJ= I c) ip if), then, by linearity, ip(t) is a solution of (3). If ip(t 0) = 0 for some to in (rl, r2) and the C j are not all zero, then by Lemma 2.2.1 we conclude that <p(t) is identically zero,

which contradicts linear independence.

2.2 Fundamental Solutions 1 9

D E FIN ITIO N : A collection <fJI (t ), . • . , <fJn(t), r l < t < r2 , of solutions of the n-dimensional first-order l inear system (3) is called a funda mental syste m of solutions of (3) if it i s l inearly independent.

The importance of a fundamental system of solutions of a l inear system is that we may describe any solution of the system in terms of the fundamental system of solutions. The problem of finding any solution then becomes one of finding n linearly independent solutions ; but even more important, we need only properties of the fundamental system in order to determine the behavior of any solution .

We now show that a fundamental system of solutions exists for equation ( 3) .

T HE O R E M 2.2 . 1 . A fundamental syste m o f so lutions of (3) exists .

Proof. Let e l , . . . , en be the linearly i ndependent set of n-dimensional vectors

jlh place e j = (0, ... , 0, 1, 0, ... , 0), j = 1, .. . , n.

(See Example (b) above.) For any t o in (r l ' r2) , let <fJI(t), ... , <fJit ) be the solutions of ( 3) satisfying <fJ ito) = e j' j = 1, . . . , n . These are all distinct, s ince each satisfies distinct initial values. Furthermore, they are l inearly independent, for if <fJ(t ) = If=1 Cj<fJj(t) == O, r l < t < r2' with the C j not all zero, then

n n <fJ(to) = L Cj<fJito) = L cjej = (cI, ... , cn) = 0.

j= I j= I

But this implies CI = . . . = Cn = 0, which is a contradiction. Therefore <fJ 1 (t ) , . . . , <fJnU) are l inearly independent, and form a fundamental system of solutions of ( 3) .

C O R O L LARY 1 . Every solut ion o f equat ion ( 3) i s a linear comb inat ion of the elements of a funda mental system of solutions .

Proof If x(t) i s a solution of (3) it is defined for r l < t < r2 ; let Xo = (x10, • • • , xnO) be its value at to. Let <fJI(t), . . . , <fJn(t ) be the

fundamental system of solutions constructed above and let cp(t) =

'Lj= 1 xjO cp /t). By l inearity, cp(t) is a solution, and furthermore <p(to) = (x10,' • • , xnO) = Xo· By uniqueness, we must have <pet) = x(t) for'1<t<'2'

Examples

(a) For the equation x = a(t)x, with aCt) a scalar continuous function on '1 < t < 'z, let cp(t) = exp[t a(t)dt ] . It is a solution, it is never zero on '1 < t < '2, and hence it is trivially independent and satisfies cp(to) = 1 . Any solution x(t) is therefore given by x (t) = Xo cp(t), '1 < t < '2' and x(to) = Xo .

(b) Consider the two-dimensional linear system x = A(t)x corresponding to the second-order equation x + x = 0, where x(t) i s an unknown scalar function. This system is given by

x=y, y = -x,

and hence

A(t)= (_� �), and all solutions (X(/) , y(t»are defined on -00 < 1 < 00. Consider the two solutions CPl(t) = (cos t, -sin I) and CP2(t) = (sin t , cos t). That they are linearly independent follows from the fact that C1 sin t + C2 cos t == 0 implies C1 = Cz = 0, (let t = 0 ; then t = nI2) . Furthermore, CPl(O) = ( 1 , 0), cpz(O) = (0, 1 ) , so any solution (x(t), y(t» is given by (x(t), yet»� = Xo CPl (t) + Yo cpz(t) for -00 < t < 00, where (x(O), yeO»� = (xo, Yo). This solution corresponds to the solution x = cp(t) of the equation x + x = 0 satisfying the initial conditions cp(O) = Xo , (p(O) = Yo .

The discussion and the theorem indicate the following: the space X of all solutions of ( 3) is a l inear space, since any linear combination of solutions is again a solution . Furthermore, X has dimension n,

since any element of the space can be described as a linear combination of the n linearly independent e lements of a fundamental system of solutions.

2.3 The Wronskian 21

Finally , we introduce the n x n matrix (t ) , whosejth column is ({J it) as constructed in Theorem 2.2.1. If we let ({J it) = «({J Ij(t), ... , ({In/t)), j = 1, .. . , n, then (t) = «((Jij(t» and (t o) = I, the identity matrix. By (t ) we mean the matrix (<pij(t», and from the above we have

C O R O L LAR Y 2. The matrix (t ) i s the solution o f the m atr ix different ial equation = A( t ) sati sfying <1>( t o) = 1. Furt her more , the solution x(t ) o f ( 3) sati sfying x(to) = Xo can be writte n x(t ) = (t )x o .

Proof Each column of<l>(t ) is a solution cpP) of (3) and hence satisfies <Pi = A(t)cp j; therefore (t) satisfies = A(t) . If Xo = (XlO, • • • , xnO), then by Corollary 1 we have

n x(t) = L xjo CPj(t) = (t)xo· j= 1

2.3 The Wronskian

We continue our discussion of the n-dimensional l inear system x = A(t) x, (4)

and will specifically determine a necessary and sufficient condition for a collection cP 1 ( t ) , . . . , CPn(t ) of solutions of (4) to be a fundamental system of solutions-that is, to be linearly independent. As in previous sections we assume that A(t) is contiouous for r1 < t < r2 •

D EF I N ITIO N : Let CPl(t ), . .. , CPn(t) be solutions of (4) where cp/t ) = (CPl / t ), ... , CPnP»· Then the scalar function (CPl �(t) ... CPl�(t}) W(t) = det : :

CPn 1 (t) CPnn( t) is called the Wronskian of CPI(t), . . . , cpnCt).

D EFINI TIO N : If CPI(t) , ... , cpit) is a fundamental system of solutions of (4), then the matrix corresponding to W(t ) is called a jimdamental ma trix.

Thus a fundamental matrix is a matrix whose columns form a fundamental system of solutions of (4) . The matrix (t ) constructed in the previous section is a fundamental matrix .

Example : For the two-dimensional system x = y, y = x, given in the previous section, the matrix «1>( t) = ( C?s t sin t)

-SIll t cos t ' - CtJ < t < CtJ,

is a fundamental matrix, and Wet ) == I, - CtJ < t < CtJ.

The following theorem shows that given any n solutions of (4) and any to in (rl' r2) , we can completely determine the corresponding Wronskian without computing the n x n determinant. Recall that for any square matrix A = (a ij) , the trace of A is given by

n tr A = L aii·

i: 1

T H E O R E M 2 .3 . 1 . (Liou ville 'sfor mula .) Let ({Jj(t ), ... , ((JnCt) be a ny n solut ion s o f equation (4) and let to be in (r l' r 2) . Then the Wron skian of ({Jt(t), . . . , ((JnCt ) is given by

W(t) = W(to) exp [( tr A(s) dS], Proo f. We will show that Wet) satisfies the differential equation

W = [tr A(t )] W, from which the conclusion follows (see Example (a), Section 2 .1). If ({J/t) = «((Jlj(t ), .. . , q>n/t»), then by the expansion by cofactors of Wet) we have

n Wet) = L ((Jij(t) I lti/t) I ,

j= 1 where I Wij(t ) I is the i, jth cofactor of Wet ) .

The cofactor I Wi/f) I does not contain the term ({Jij(t) , so if we regard Was a function of the ({Jij we have aWjo({Jij = I Wijl and , by the chain rule,

• n oW n ( n ) W = L - <Pij = L L <Pij Iltijl .

i,j= 10({Jij i:l j= 1

2.3 The Wronskian 23

Let "IfI be the matrix corresponding to Wet), differentiate its ith row, and call the corresponding determinant Wi(t). The expression in parentheses in the above equation is the expansion of Wi(t) by cofactors, and therefore Wet) = I7= I Wi(t).

Each column «{J1j(t), . .. , ((In/t)) of"lfl satisfies (4) and therefore

<Pij = I�=I aik(t)({Jkj for i = 1, . . . , n) hence

({Jll ({Jln ({J11 ({J In

n n fti(t) = det <Pit <Pin = det Iaik({Jkl I aik ({Jkn k=1 k=1

({Jnl ({Jnn ({In I ({Jnn

Multiply the kth row for k #- i of the last matrix given by - aiit) and then add it to the ith row. This does not change the value of the determinant but gives the relation

fti( t) = det aii ({Jil aii ({Jin = aii(t)W(t).

It follows that Wet) = I7= I aii(t) Wet) = [tr A(t)] Wet), which leads to the desired result.

Since exp [fr� tr A(s) dS] is never zero, the theorem implies that

the Wronskian of any collection of n solutions of (4) is identically zero- W(to) = 0 for some to-or never zero on (rl' r2). The latter case characterizes a fundamental system as the following theorem shows.

THEOREM 2.3.2. A necessary and sufficient condit ion for ({JI(t), ... , ((In(t) to be a fundamental system of solutions of (4) is that

Wet) #- 0 for r1 < t < r2 •

Proof Let <p \ (t) , . . . , <Pn(t) be a fundamental system of solutions of (4) and let <p(t) be any nontrivial solution. By Corollary I of Theorem 2.2.2 there exist C \ , . . . , Cn not al l zero such that <p(t) = I�= 1 Cj <pj(t) , and by the uniqueness of the solutions the Cj are unique. If C = (ct , . . . , cn) and (t) i s the fundamental matrix of (t)c. For any t i n (r1 ' r2) this is a system of n l inear equations in the unknowns C1 , • • • , Cn , which has a unique solution, and this implies det (t) = W(t) =1= o.

Conversely, if Wet) =1= 0 for rl < t < r2 , this implies that the columns (t) are l inearly independent for rl < t < r2 • Since they are solutions of (4), they form a fundamental system of solutions.

Finally, equipped with the above two results we are able to state the following sharper version of Corollary 2 of Theorem 2.2. 1 .

C O R O L L A R Y . Any fundamental matrix (t) is the solution of the matrix differential equation = A(t). Furthermore, the solution x(t ) of (4) satisfying x(to) = Xo can be written x(t) = (t)- I (tO)XO , and the matrix net) = (t) - I (tO) is a fundamental matrix satisfying O(to) = I.

Proof We have left only to prove the second statement. Given the fundamental matrix (t), the solution x(t ) satisfies x(t) = (t)c for some constant vector c, and therefore Xo = (to)c. But W(to) = det (to) =1= 0 implies that <1> - 1 (10) exists, and thus c = - l(tO)XO ' which gives the first result.

Furthermore, det - I (tO) = W- I(to) =1= 0, and this implies that the columns of net) = (t)- I (/o) are linea�ly independent, since those of <1>(/) were. The columns of O(t) are l inear combinations of those of (t) and are therefore solutions of (4) . Hence O(t) i s a fundamental matrix and O(to) = I.

Example: Consider the two-dimensional system x = 2x + y, and hence

A(t) = G !) , y = 3x + 4y,

2. 4 The Nonhomogeneous Linear Equation 25

and tr A(t) = 2 + 4 = 6. All solutions are defined on - 00 < t < 00 , so if we let to = 0 the Wronskian of any fundamental system of solutions is

W(t} = w(o) exp [{ 6 dS] = W(0)e6t •

A fundamental system of solutions is given by

CP t(t) = (x(t) , yet)) = (et, _ et) and

CP2(t) = (eSt , 3eSt) , Therefore ( et <1>(t) = t - e and hence W(O) = det <1>(0) = 4. The Wronskian is therefore Wet) = 4e6 t, as may be verified by direct calculation.

Finally, we have

and hence

- t) 1 ' "4

is a fundamental matrix satisfying nco) = I. Therefore any solution cp(t) can be given by

cp(t) = n(t)cp(O) .

2.4 The Nonhomogeneous Linear Equation

In this section we introduce the method of variation of parameters and use it to obtain the solution of the first-order n-dimensional nonhomogeneous equation

x = A(t)x + B(t) . (5)

26 The Linear Equation : General Discussion

Here, as before, A(t ) = (a ij( t» , i, j = 1 , . . . , n, and B(t) = (b l (t) . . . , bit» , are assumed to be continuous on rl < t < r2 • By our previous discussion this implies that solutions exist and are unique in

B = {(t, x) I r l < t < r2 ' - 00 < Xi < 00, i = 1 , . . . , n } ,

and every such solution is defined on rl < t < r2 • In the previous section we obtained the result that the solution

x(t) of X = A(t)x satisfying x(to) = Xo is given by x(t ) = <1>(t)xo , where <1>( t) i s a fundamental matrix satisfying <1>( to) = I. To apply the method of variation of parameters, we assume the solution x( t ) of (5) satisfying x(to) = Xo can be given by x(t ) = <1>(t )e(t ) , where e(t ) = (c l ( t ) , . . . , cn(t » . This leads to

TH E O R E M 2.4. 1 . The solution x(t) of equation (5) sati sfying x(to) = xo , r l < to < r2 , is gil'en by

t x(t) = <1>(t)xo + I <1>(t)<1>- I (s)B(s) ds,

to

where <1>(t) is the fundamental matrix of the equation x = A(t)x sat isfying <1>( to) = I.

Proof The representation xU) = <1>(t)c(t) , c(t) = (cl ( t) , . . . , enU» is val id if and only if ( i ) c( to) = xo , since <1>( to) = I, and ( i i) <1>(t)c = B(t ) . This last result is obtained by substituting x(t ) = <1>(t)c(t) into (5) . By the product rule for differentiation this gives

x = <bc + <1>c = A(t)<1>c + B(t) .

But <1>(t ) i s a fundamental matrix of x = A(t )x and hence satisfies the relation , which implies that $(t)c = B(t) , and conversely.

Furthermore, W(t) = det $(t) =I 0 for r l < t < r2 , and therefore $ - I ( t ) exists on (r l , r2) ' The relations ( i ) and ( i i ) are then equivalent to

But the solution of this equation is g iven by t

e(t) = Xo + J $ - I (s)B(s) ds , 1'1 < t < 1'2 to

and this gives the desired result .

2.4 The Nonhomogeneous Linear Equation 27

It should be remarked that the use of the representation above to obtain explicit solutions when n ;:::: 3 i s very l imited . For it involves finding a fundamental matrix (hence a fundamental system of solutions of x = A (t )x), then computing an inverse matrix , and finally performing the indicated integration . Finding a fundamental matrix even when n = 2 may prove to be a very difficult task.

The value of the representation i s that , even knowing only properties of the fundamental matrix <1>(t ) and the behavior of B(t) , we may be able to obtain considerable information about the solution x( t) . This will become especially clear in Chapter 5 , in which stabi lity of solutions of equations of the form above are discussed .

A much simpler representation of solutions of (5), not involving <l> - I (t) , can be given in the case A(t) = A, a constant matrix, and we assume to = O. However, we need the fol lowing lemma.

L E M M A 2.4. 1 . If<l>(t) is the fundamental matrix of x = Ax, with A a constant matrix and <1>(0) = I, then <l>(t - IX) = <1>(t)<1> - I (IX) for every IX .

Proof Given the real number IX , le t 0, ( t ) = <1>(t)<l> - I (IX) ; since <l>(t) satisfies the equation , it fol lows that QI (t) is the solution of 0 = A Q with in itial condition Q(IX) = I. But Q2(t) = <l>(t - IX) satisfies Q2(ex) = <1>(0) = I, and O2 = A<1>(t - IX) = A Q2 . By uniqueness, Q2(t) = Q, (t) .

Note that even for a specific value of IX the above result wil l not hold for A = A(t) un less A(t - IX) = A(t )-that is, that IX is a period of A(t) . Finally, the above two results lead to the following representation for solution of

x = Ax + B(t) , (6)

where A = (a ij) i s a constant matrix and B(t ) i s continuous on r , < t < r2 • We assume that r l < 0 < r 2 •

T H EO R E M 2.4 .2 . The solution x( t ) of equation (6) satisfying

x(O) = Xo for r l < 0 < /"2 is given by

I x( t ) = <1>(t)xo + f <l>( t - s)B(s) ds ,

o

where (t) is the fundamental matrix of the equation x = Ax satisfying <1>(0) = I.

Example: Consider the two-dimensional system x = 2x + y + cos t, y = 3x + 4y + t . Hence

A(t) = A = G �) and

so all solutions exist and are uniquely defined on - 00 < t < 00 .

The solution <p(t) = (x(t), y(t» satisfying <p(0) = (xo , Y o) i s given by

G�:n = (t)G:) + f� (t - S)eosS s) ds,

where

as given in the last example of the previous section . The reader may verify, for instance, that the solution satisfying <p(0) = ( 1 , 1) is given by <pet) = (x(t), yet»� , where

5 I 1451 51 5 . 1 1 1 6 x( t) = 8 e + 2600 e + 13 sm t - 26 cos t + 5" t + 25 '

5 I 4353 51 9 . 3 2 7 yet) = - - e + -- e - - sm t + - cos t - - t - - .

8 2600 26 1 3 5 25 Even the simplest problems lead to rather laborious calculations .

2.5 The nth-Order Linear Equation

We wi l l n o w apply the results obtained in the previous sections to a consideration of the nth-order l inear equation

in) + a \ (t)in - 1 l + . . . + ai t)y = O. (7)

Here y = y(t) is an unknown scalar function, and the a;(t), i = I , . . . , n, are continuous on rJ < t < rz . The equation is called homogeneous since the right side is zero.

Using the substitution given in the first chapter we let Xl = y and Xz = y, . . . , X

n = in- l ) . Then (7) is transformed into the first-order

n-dimensional linear system x = A(t)x,

where

A(t) =

o o

o - ait)

1 0 o 1

o o

(8)

Here X = x(t) = (x l (t), . . . , xn(t» = (y(t) , . . . , in - 1 )(t» is an unknown vector function. Also note that for (8) the initial condition x(to) = Xo is equivalent to an initial condition for (7) of the form

( ) . ( ) . (n - I )( ) ( n- I ) y to = Yo , Y to = Yo , . . . , Y to = Yo , (9)

where Yo , Yo , . . . , Ybn- I ) are given constants. This is the form of in i tial conditions for the equat ion (7) .

Equation s (7) and (8) are equivalent, since to a sol ution y = ljJ(t)

of (7) there corresponds the solut ion

X = cp(t) = (ljJ(t) , tiJ(t) , . . . , ljJ(n- 1 ) (t» of (8) . Conversely, g iven a so lution X = cp(t) = (CP I ( t) , . . . , CPnCt» of (8), there corresponds the solut ion y = cP I ( t) of (7), and (8) implies i' = CP I = <Pz , . · · , in - I ) = CPn - 1 = <Pn · We may conclude that given to in (r l ' r2) and constants Yo , Yo , . . . , y&n - J ) , there ex ists a unique so lution y = y(t) of (7) , wh ich i s defined on 1" 1 < t < r2 and satisfies the in itia l cond it ions (9).

Tn view of the equivalence between equations ( 7) and (8), the

discussion of fundamental systems of solut ions of (7) and the correspond ing Wronskian is considerably s implified . This i s one of the great advantages of first having discussed system of the form (8) .

D E F I N I T I O N : A collection

of solutions of (7) is called aJundamental system oj solutions of (7) if i t is l inearly independent. We can now prove immediately T H E O R E M 2 .5 . 1 . A Jundamental system oj solutions oj equation (7) exists.

Proof By Theorem 2.2. I a fundamental system of solutions of (8) exists ; for example, lP l (t ) , . . . , lPit ) , where lPit) = (lP l j( t), . . . , lPn/l» . Furthermore, given to in (rl , '2) we may assume

jib place lPito) = (0, . . . , 0, 1 , 0, . . . , 0) = ej , j = 1 , . . . , n.

By the correspondence of solutions of (7) and (8) we have lP/t) = (yit), ,Pit), . . . , y�- I )(t»

for some solution y = Yj(t) of (7) . The collection Yl(t), . . . , Yit) are distinct nontrivial solutions, since they satisfy distinct initial conditions and Yj(t) == 0 for ' 1 < t < '2 would imply that epj(t) == 0, which is impossible.

Finally, if there existed constants Cl > • • • , Cn not all zero such that L)= 1 cjy/t) == 0 for 'I < t < '2 ' then

n n L cj ,P/ t) == 0, . . . , I cj yjn - I )( t) == 0, j = 1 j = 1

This implies that n L Cj lP/t) == 0,

j = 1 which contradicts the fact that lP l (t), . . . , lPn(t) is a fundamental system of (8).

C O R O L L A R Y. Given any solution y = y(t) oj (7) and a Jundamental system oj solutions Yl (t) , . . . , Yn(t) oj (7), there exist constants c 1 , • • • , Cn such that

n yet) = L Cj y/ t) , j = 1

2.5 The nth-Order Linear Equation 3 1

Proof Given to i n (r1 , r2), suppose that y(to) = Yo , y(to) = . (n - l )( ) (n - l ) Th .. ( ) b 1 Yo , . . . , Y to = Yo . erelore C1 , · . . , Cn = C must e a so u-tion of the system of equations

n Yo = L cj y/to) , j = 1

n n • " • ( ) (n - 1 ) '\' (n - 1 )( ) YO = L. Cj Yj tO ' ' ' · ' YO = L. Cj Yj to · j = 1 j = 1 The matrix of coefficients of this system i s

(Yl ( to) Yl( tO)

( to) = .

y\n� 1 )( to) But (t) for r1 < t < r2 is a fundamental matrix, since its columns are a linearly independent set of solutions of (8) ; hence - I (tO) exists. If Yo is the vector (Yo , Yo , . . . , Y6n-1 » , then the solution is given by

C = - I ( tO) Yo .

(7). We now define the Wronskian of a collection of n solutions of

D EF I N I TI O N : Given any collection YI (t) , . . . , YnCt) of solutions of (7), then

(Y I YI

Wet) = det : y

\n-I ) y

�n- I )

is called the Wronskian of Y I (t) , . . . , Yn(t) · As before, if the YI ( t ) , . . . , Yn( t ) are a fundamental system of (7),

then the matri x corresponding to W(t) is cal led a fundamental matrix. In any case, note that the columns of the matrix corresponding to w(t) are 11 solut ions of the system (8) . We may therefore state immed iately a result analogous to Theorem 2.3 . 1 , noting that tr A(t) = - a j (t) .

T H E OR E M 2.5 .2 . The Wronskian Wet) of any collection Yl (t), . . . , Y,,(t) of solutions of equation (7) satisfies the relation

Finally, we have the result corresponding to Theorem 2.3.2 ; the proof is virtually the same.

T H E O R E M 2.5 .3 . A necessary and sufficient condition for Yl (t), . . . , Yit) to be afundamental system of solutions of equation (7) is that

Wet) i= 0,

Examples

(a) Given the second-order equation ji + a(l)y = 0, then for any two solutions YI( f) and 12(t) we have

Wet) = det (� I ( t) �2(t») = constant, r 1 < t < r2 . Y l ( t) Y2(t)

The constant i s nonzero if and only if Yl and Y2 are linearly independent.

(b) Given the third-order equation

3 2 2 y(3) + _ ji _ _ y + _ Y = ° t t2 t3 ' t > 0,

then a l (t) = 3/t. The Wronskian of any three solutions Yl (1), 12(1), 13(t) satisfies

Wet) = W(to) exp l- ( � dS] = W(to) e;) 3 ,

where t, to > O. A fundamental system of solutions is given by Yl (l) = t ,

12(t) = t log t , and Y3(1) = 1 / 1 2 . Therefore, if to = 1 , we have

2.6 The Nonhomogeneous nth-Order Linear Equation 33

t log t 1 t2

W(t) = det 1 1 + log t 2 and W(1) = 9, - (3

1 6 0 (4

so Wet) = 9/t 3, which may be verified directly. (c) The fact that linear independence implies a nonvanishing

Wronskian is a property of solutions of linear equations. The functions y,(t) = ( 3 and Y2(t) = I t l 3 are l inearly independent on - 00 < I < 00, but

( t3 I t 1 3 ) W(t) = det 3t2 3t l t l =0,

- oo < t < oo .

Evidently Yl (t) and Y2(t) could not both be solutions near 1 = 0 of a second-order linear equation, since they both satisfy yeO) = yeO) = 0, yet are distinct. This would violate uniqueness (see Problem 7 of this chapter) .

Note that no general methods exist for finding fundamental systems or even one solution of the nth-order linear equation. Howe\'er, a method using power series is available for a rather large class of second-order linear equations ; it is discussed in Appendix A .

2.6 The Nonhomogeneous nth-Order Linear Equation

To conclude this chapter we will discuss the nth-order linear nonhomogeneous equation

y<n) + a , (t)y<n - ' ) + . . . + ait)y = bet) , (10)

and will use the method of variation of parameters to obtain its solution. Here y = y(t) is an unknown scalar function, and a/I), i = 1, . . . , n, and b(t) are continuous on r1 < t < r2 .

The substitution given in the previous section transforms ( 1 0) into the first-order n-dimensional system

x = A(t)x + B(t), ( 1 1)

where A(t) is given in equation (8) and B(t) = (0, . . . , 0, b(t» . We may conclude that solutions of ( 1 0) satisfying initial conditions of the form

y( to) = Yo , y( to) = Yo , . . . , y(n - 1 )( to) = y�n - 1 ) , ( 12) where rl < to < r2 , exist and are uniquely defined on rl < t < r2 •

To employ the method of variation of parameters, let YI (t), . . . , Y

n(t ) be a fundamental system of solutions of the homo

geneous equation corresponding to ( 1 0) . Then assume that a solution yet) of ( 1 0) satisfying in itial conditions ( 1 2) can be expressed as

n

yet) = L c/t)y/t), (13) j = l

where cl (t) , . . . , cnCt) are to be determined . This leads to

THE O R E M 2 .6 . 1 . The solution of equation ( 1 0) satisfying initial conditions ( 1 2) is given by

n f' b(s) W(s) ye t) = q>(t) + W(tO) - 1 b/it)

, [ s ) ] ds ,

) - 0 exp - f a l (u) du ' 0

where

(i) q>(t) is the solution satisfying initial conditions ( 1 2) of the corresponding homogeneous equation,

(ii) YI(t), . . . , YnCt) are a fundamental system of solutions of the homogeneous equation and W(t) is their Wronskian, and

(iii) Uj(t) is the determinant obtained from W(t) by replacing the jth column by (0, . . . , 0, I) .

Proof The matrix <1>(t) withjth column (yP) , yit), . . . , yCn - l )( t» is a fundamental matrix of x = A(t)x ; therefore we assume that c(t) = (c l (t) , . . . , cit» can be chosen so that yet) = <1>(t)c(t) is a solution of ( 1 1 ) . This is equivalent to assuming that c(t) can be chosen so that we have

n n yet) = L c/t)y/t), ye t) = L c/t)y/t) , . . . , j = l j = l

n y<n - I j e t) = L cj( t)yjn - I je t) , j = l

3 5

and Y(t ) = (y(t ) , y(t) , . . . , y<n - l )(t » satisfies Y(to) = (yo , yo , . . . , ybn - 1 ) . But now we may proceed as in Theorem 2.4. 1 to show that this

is equivalent to the relation (t)c = B(t) . Hence

c( t) = Co + { - t (S)B(S) ds, 10

so that I

yet) = (t)co + I (t)- t (s)B(s) ds, 1 0

where Co = (c I O ' . . . , CnO) = c(to) · The expression ( t)co i s a solution of the homogeneous system

corresponding to ( 1 1 ) , so we may write

(t)Co = (cp( t) , cj>( t), . . . , cp(n - l je t»�

= ttl cjO y/t), jt tCjO y/t), . . . 'jtt

CjO yjn - l )( t») .

The relation Y(to) = (Yo , Yo , · · · , Ybn - 1 » implies that cp(t) = LJ= l Cjo y/t ) i s a solution of the homogeneous equation corresponding to ( 1 0) satisfying the init ial conditions ( 1 2) .

To determine cj( t ) for j = I , . . . , n explicitly, we note that (t)c = B(t) gives the system of equations

+ . . . + ynC t)cn = 0,

y\n - Z )( t)C t + . . . + y�n - Z )( t)cn = 0,

y\n - l )(t)c 1 + . . . + y�n - I )(t)cn = be t) .


The determinant of the coefficients of C'I ' . . . , Cn is W(t), the Wronskian of the fundamental system Yl(t), . . . , ynCt) . Using Cramer's rule and the form of the Wronskian given in Theorem 2 .5 .2, we have

. ( ) _ b(t)Wj( t) c · t - ----::----"-,------::

J w(to) exp [ _ ( a t (S) ds] '

and hence - 1 I t b(s)Wj(S) cit) = CjO + W(to)

[s ] ds .

to exp - I a l(u) du to Finally, the relation n

yet) = L cj(t)y/t) j= l

gives the desired result. Example : For the second-order equation ji + a l (t)y + a2(t)y = bet) we have Wl(t) = - Y2(t), W2(t) = Yl (t) for any fundamental pair of solutions Yt (t) and Yz(t) of the homogeneous equation. If Wet) i s their Wronskian, then the solution yet) satisfying y(to) = Yo , y(to) = Yo is given by

It b(s)yz(s) yet) = C IY t(t) + C2 Y2( t) - Y t ( t) ds to W(s)

+ (t) It b(S)Y l (S) d Y2 to W(s) s,

where Ct and C2 are chosen so that cp(t) = cIYt (t) + C2 Y2(t) satisfies the initial conditions.

For instance, the functions Yt (t ) = t 1 /2 and Yz(t) = t 1 /

2 log t are a fundamental system of solutions of the equation

1 ji + 4t2 Y = 0, t > 0,

Problems 37

and Wet) = 1 . Using the expression above and letting to = I , we may verify that the solution y(t) satisfying y( l ) = I , y( l ) = 3/2 of

1 .. + _ _ t3 /2 Y 4t2 Y - , t > 0,

is given by

yet) = !t l /2 + 1t l /2 log t + tt7 /2 .

Problems

1. Given the first-order l inear equation

x = a(t)x + bet),

where x = x(t) is an unknown scalar function and aCt), bet) are con

tinuous on rl < t < r2 , show that the solution satisfying x(to) = Xo is

given by

x(t) = expU: a(s) dS] {xo + ( b(s) exp [ - ( a(u) dU] dS} . Use this to find solutions of the equations

(a) x = x + 2 e' , x(O) = I .

(b) x = - r lx - r 2, x(1) = I .

(c) x = -(t + l)t- 1x - 3t2e- ' , x(l ) = 1 .

2. Verify that cp(t) is a solution and then apply the results of the previous problem and of Problem 8, Chapter 1 , to find a solution x(t) satisfying

x(to) = Xo of the following Riccati equations.

(a) x = x + 2r 3x2 - t2 , cp(t) = 12.

(b) x = x2 + cos2 t + cos 1 - 1 , cp(t) = sin I. In (b), discuss the behavior of the solution as 1 approaches ± 00 .

3. Given the inhomogeneous first-order system

x = 3x - y + 1 ,

y = 4x - y + I,

show that

<Pl(t) = (e', 2e'), <PZ(t) = (te', -e' + 2te'),

is a fundamental system of solutions of the corresponding homo

geneous system. Then find the solution <pet) of the given system

satisfying <p(o) = ( l , 0).

4. Given the nth-order l inear equation and a solution Y = Yl(t) , show

that the substitution Y = Yl (t) J ' u(s) ds results in a linear equation

of order n - 1 in u = u(t). This is the method of reduction of order. Use this method to find a second linearly independent solution

of the following equations.

(a) Y + 4ty + (4tZ + 2)y = 0, Yl(t) = e- ' z .

(b) Y - (2 secz t)y = 0, Yl(t) = tan t.

t + 1 2(t - 1 ) (c) Y - -t - Y - t

Y = 0, Y l(t) = eZ ' .

5 . (a) Given the equation

4 2 sin t Y + - y + - y = -t tZ t '

t > 0,

show that Yl (t) = l /t and Yz(t) = l /tZ are a fundamental system of solutions of the corresponding homogeneous equation. Then find the

solution yet) of the given equation satisfying y(l ) = 1 , y( l ) = 0.

(b) Proceed as in (a) for the equation

2 Y - t Y + Y = t z , t > 0,

where Y I (t) = sin t - t cos t, yz(t) = cos t + t sin t, and the solution yet) must satisfy Y(7T/2) = 0, Y(7T/2) = 1 .

6. Show that i f XI (t) is a nontrivial solution of x + a(t)x = 0, where x = x(t) is an unknown scalar function and aCt) is continuous on

/" 1 < t < "z , then

X2(t) = XI (t) r [XI (S)] - 2 ds ' 0

is another l inearly independent solution .

Problems 39 7. Given scalar functions X1(t) , X2(t), continuously differentiable and

linearly independent on a < t < b, show that if their Wronskian is identically zero on (a, b), then there exists to in (a, b) such that

X1(tO) = X2(tO) = X1(tO) = X2(tO) = 0.

8. Given scalar functions U1 (t), • . . , u.(t), continuous on a � t � b, show that they are linearly dependent if and only if det A = 0, where A = (au) and

. b aij = J a ui(t)uit) dt, i, j = 1 , . . . , n.

9. Using the technique described in Appendix A for the case s i= 0, positive integer, find two l inearly independent solutions of the following equations near the regular singular point Zo = 0.

d2w dw (a) 2Z2 - - Z - + (Z2 + I)w = 0. dz2 dz d2w dw (b) 2Z2 - - Z - + (1 -Z2)W = O. dz2 dz

d2w dw (c) Z2 dz2 + z dz + (Z2 - 0( 2)W = 0, 0( i= 0, positive integer (Bessel's

equation).

10. The following equations are examples of the case s = positive integer, nonlogarithmic case. Find two linearly independent solutions near the regular singular or ordinary point Zo = 0.

d2w dw (a) z dz2 + 2 dz + Z2W = 0.

d2w dw (b) Z dz2 + (z - I ) dz - w = 0 .

d2w dw (c) (1 - Z2) - - 2z - + 0« 0( + l )w = 0, 0( constant (Legendre's dz2 dz

equat ion) .

1 1 . The following equations are examples of the case s = 0, logarithmic case. Find two l inearly independent solutions near the regular singular point Zo = 0.

40

d2w dw (a) z

dz2 + dz - zw = O.

d2w dw (b) Z dz2 + dz

+ Z2W = O.

d2w dw

The Linear Equation : General Discussion

(c) Z2 dz2

+ 3z dz + (l + z)w = O.

12. Using the results of Appendix B, show that if Hill's equation with T = 7T has a nontrivial periodic solution with period n7T for n > 2, but no solution with period 7T or 27T, then all solutions are periodic with period n7T.

13. Consider the equation

x + Ep(t)X = 0,

where pet) is real-valued, continuous, and periodic with period T, and E is a parameter. (a) Show that the fundamental pair of solutions XI(t, E), xi!, E) satisfying the initial conditions

XI(O, E) = 1 , X2(0, E) = 0, X2(0, E) = 1

can be formally expressed by the power series

XI (t, E) = 1 + EII(t) + E2/2(t) + . . . , X2(t, E) = t + EgI(t) + E2git) + . . . ,

where I. and g. satisfy the relations 1.(0) = g.(O) = U.(O) = 1.(0) = 0 and

J,. + P(t)f. - l = 0, g. + P(t)g. - l = 0,

for all n, where lo(t) = 1 , go(t) = t. (b) Assuming the expansion in (a) is valid and letting E = 1 and T = 7T, show that the condition pet) < 0 implies that

X I (7T, 1) + xi7T, 1) > 2.

Show that this implies that the corresponding Hill's equation possesses unbounded solutions.

14. Given the first-order linear equation of Problem 1 , with aCt) and bet) periodic of period T, show that it has a periodic solution if and only if exp[J� a(s) ds] #- 1 . Find the periodic solution of the equation x = tx + sin t.

C H A P T E R

3 The Linear Equation

with Constant Coefficients

3.1 The nth-Order Linear Equation

In this section we will discuss the equation z(n) + a l z(n - I ) + . . . + anz = 0, ( 1 )

where z = z(t) is an unknown scalar function, possibly complexvalued, and ai ' . . . , an are real or complex constants. The importance of this equation is twofold. First of all, we can immediately determine a fundamental system of solutions, and hence any solution . This is extremely difficult for the general l inear equation, whereas for ( 1 ) it merely involves the algebraic process of finding the roots of an associated polynomial.

Second, and more important, a large number of physical phenomena can be described in terms of an equation of the form ( 1 ) o r b y a convenient " linearization ." For example, the motion o f a simple pendulum is governed by the equation (j + k sin e = 0, where e = e(t) is the angle of displacement and k i s a constant. We " linearize " the equation by requiring that l e i be small, and consider only the equation (j + ke = 0, which is of the form ( l ) above and for which solutions can be explicitly given .

The question arises, for example, whether the solution eCt) of the l inear equation adequately describes the motion of a pendulum.

42 The Linear Equation with Constant Coefficients

If we rewrite the pendulum equation as e + kO + e(O) = 0, we are asking the question, " If l e(O) 1 is small for 1 0 1 small, can solutions be described locally in terms of the behavior of solutions of the l inear equation ? " This type of question will be discussed at length i n Chapters 4 and 5 , and i t will b e advantageous to know beforehand about l inear equations with constant coefficients.

Equation ( 1 ) i s a special case of the l inear equation discussed i n Section 2 .5 . Therefore, given the in itial conditions

z(to) = Zo , z( to) = zo , . . . , z (n - l )( to) = z�n - l ),

we know that a unique solution of ( I ) exists which satisfies them and, since ai ' . . . , an are constants, the solution is defined on - 00 < t < 00 . Furthermore, the results proved i n Section 2 .5 apply here, s o we may state immediately the following theorems.

T H E O R E M 3 . 1 . 1 . A funda mental syste m of solut ions of equat ion ( 1 ) ex ists, and every solut ion can be expressed as a l inear co mb inat ion of the ele ments of a fundamental system .

T H E O R E M 3 . 1 .2 . The Wronskian Wet) of any collect ion ZI ( t), . . . , zn(t) of solut ions of equat ion ( 1 ) sat isfies the relat ion

Wet) = W(0)e - a 1 t , - 00 < t < 00 ,

andfor such a collect ion to be afunda mental syste m of solutions a necessary and suffic ient condit ion is that W(t) i= 0, - 00 < t < 00 .

T o determine explicitly the form of a fundamental system of solutions we will require a few prel iminary lemmas and remarks . Recall , first of all, that for the complex number z = u + iv the exponential function eZ is defined by

eZ = eu + iv = eU(cos v + i s i n v),

and the usual laws of exponents hold. Furthermore, eZ = 1 if and only if z = 0.

LEM M A 3 . 1 . 1 . rr ), is any real or complex number, then

d , . _ (eAt) = AeAt . dt

3. 1 The nth-Order Linear Equation 43

Proof It is certainly true for real A., so we first consider the purely imaginary case A. = iv. Then

and

eAt = eivt = cos vI + i sin vt

d dl

(eAt) = - v sin vI + iv cos vI

= iv(cos vI + i sin vt) = ,.leu. Finally, in the case A. = u + iv,

and the conclusion follows from the product rule for differentiation. To simplify the discussion we will use operator notation as

follows : given any complex or real-valued scalar function z(t), we define the differentiation operator D recursively by

Dz = z,

where n is any positive integer. By the l inear properties of differentiation we may express equation ( 1 ) as

Dnz + a l Dn - lz + . . . + an z = L(D)z = 0,

where L(D) is the formal polynomial operator given by L(D) = Dn + a1 Dn - 1 + . . . + an '

(2)

(3) Again by the linear properties of differentiation it is evident that given any such polynomial operator L(D) and sufficiently differentiable functions Z l ( t ) and zz ( t ) , we have for any constants a and b

L( D)(az l + bzz ) = aL( D)zl + bL( D)zz .

Furthermore, given another polynomial operator M(D), then we may define a sum and product by

(L + M )( D)z = L( D)z + M (D)z,

(LM)( D)z = L( D)(M (D)z),

where z = z(t) is any sufficiently differentiable function.


Examples

(a) If L(D) = D2 + 1 and M(D) = D3 + 2D2 + 1 , then (L + M)D = D3 + 3 D2 + 2 and (LM)(D) = D5 + 2D4 + D3 + 3 D2 + 1 = (ML)(D).

(b) The above multiplication is not commutative in the case of nonconstant coefficient polynomial operators. For example, if L(D) = t 2 D and M(D) = D, then

(LM)(D)z = t2 D(Dz) = t2 D2Z,

whereas (ML)(D)z = D(t2 Dz) = 2tDz + t2 D2Z.

D E F I N I T I O N : Given equation ( 1 ) , the polynomial L(P) = pn + a1pn - l + . . . + an

i s called the characteristic polynomial of ( 1 ) .

The characteristic polynomial of ( 1 ) is the key to finding a system of fundamental solutions, as the following theorem and its corollary show.

T H E O R E M 3 . 1 . 3 . If L(p) is any arbitrary polynomial, then L(D)eAr = L(;t)eA r for any real or complex number ;t.

Proof By the definition of the differentiation operator and the result of Lemma 3 . 1 . 1 , we have for any positive integer k

D\ eAr) = Dk - 1 (DeAr) = Dk - 1 (;teA t)

= Dk - 2(D;teAt) = . . . = ;tkeAt.

If L(D) = IJ=o aj Dn - j, where we define DO by DOz = z, then n n

L(D)elt = L ajDn - j(elt) = L ajAn - jeAt = L(A)eA t . j = O j = O

C O R O L L A R Y . The function z(t) == eAt is a solution of equation ( l )

if and only if ;t is a root of L(p), the characteristic polynomial of ( 1 ) .

3. 1 The nth-Order Linear Equation

Proof

L(D)z(t) = L(D)eM = L(A)eM,

45

- 00 < t < 00 ,

and the right side i s identically zero i f and only if L(A) = 0 ; that is, A is a root of L(P).

Example: For the equation

z(3 ) - 4£ + 61 - 4z = 0, the characteristic polynomial is

L(P) = p3 - 4p2 + 6p - 4,

whose roots are p = 2, 1 ± i. Therefore the functions

e2 t, e( l ± i) t = et(cos t ± i sin t)

are solutions.

Since L(p) is a polynomial of degree n, it has n roots A I ' . . . , A.n , so we can obtain n solutions of ( I )-namely, Zj(t) = eljt, j = I , . . . , n. Using these solutions we will obtain a fundamental system of solutions of ( 1 ) . We will first consider the case in which the roots Aj , j = I , . . . , n, are distinct.

T H E O R E M 3 . 1 .4. If the characteristic polynomial L(p) of equation ( 1 ) has distinct roots Aj , j = I , . . . , n , then the functions zit) = eAj t, j = I , . . . , n, are a fundamental system of solutions of ( 1 ). Proof The functions zP) for j = I , . . . , n are solutions by the

previous corollary, so we need only prove that they are linearly independent. The simplest way to prove this is to construct their Wronskian W(t) and note that

W(O) = det

1 1 A I An

This is Vandermonde's determinant and is never zero if the AI ' . . . , An are distinct. From Theorem 3 . 1 .2 it follows that W(t) i= ° for - 00 < t < 00 ; hence we have a fundamental system of solutions.

An alternate proof is indicated as follows : suppose L:J= o cj e).jt = 0, - 00 < t < 00 , C1 , . . . , Cn not al l zero, and we assume C1 i= 0. If we multiply the sum by e - ). l t and differentiate, we obtain

n L: (Aj - A 1 )cj e().r ).dt = 0, j = 2

- oo < t < oo .

If Ck is the first nonzero Cj , multiply by e - Uk - ).dt , differentiate, and so on. After a finite number of steps we obtain the relation A cs e().s - ).r ) t = O, where Cs is the last nonzero Cj , r < s � n, and the constant A depends only on the differences A j - )'j for i i= j and is nonzero . Therefore C s = 0, and we procede as before to show that c2 , • • • , Cs- 1 = 0, and finally that c t e). t t = 0, - 00 < t < 00 , implies C1 = 0. We conclude that zit ) = e).jt, j = 1 , . . . , n, are l inearly independent.

In the case where the coefficients aj of L(p) are real, then complex roots of L(p) occur in conjugate pairs. If ). = U + iv is a root, so is X = U - iv, and the corresponding solutions are

ZI (t) = e( u+ i v ) t = eut(cos rt + i s in vt)

and (4) Z2 (t) = e(u - jv ) t = eut(cos rt - i sin vt ) .

Since a l i near combination of solutions of ( I ) i s also a solution, this implies that the real-valued functions

(5) are solutions. The fol lowing theorem shows that when the coefficients of L(p) are real and the roots are d istinct, a fundamental system of real-valued solutions exists.

T H EO R E M 3 . 1 . 5 . Suppose the characteristic polynomial L(p) of equation ( I ) has real coefficients, distinct real roots )'j ' j = I , . . . , k, and distinct pairs of conjugate complex roots Aj = uj + iVj ' Aj + J = Uj - iVj , j = k + I , k + 3, . . . , n - I . Then the real-wlued functions Zj(t) = e).jl , .i = I , . . . , k, Zj(t ) = eUj t cos v/' Zj + t ( t ) = e Uj t sin vit) , .i = k + I , k + 3, . . . , n - I , are a fundamental system of solutions of ( I ) .

3. 1 The nth-Order Li near Equation 47

Proof By the preceding theorem and the remarks above the functions represent n solutions of ( 1 ) . Suppose a nontrivial linear combination of these solutions was identical ly zero on - 00 < t < 00 . By (3) and (4) , we have

and hence the l inear combination is equivalent to an expression of the form Ii= l cj eAjt = 0, - 00 < t < 00 . From the foregoing argument, it fo l lows that C I , • • • , Cn = 0, and this implies that the coefficients of the origina l l inear combination are al l zero. Therefore Z I (1) , . . . , zit) are l inearly independent.

Examples

(a) For the equation Z( 4 ) - 4Z( 3 ) + 1 2z + 4i - 1 3z = 0, the characteristic polynomial is L(p) = p4 - 4p3 + 1 2p2 + 4p - 1 3 ,

whose roots are p = ± I , 2 ± 3i. A fundamental system of solut ions i s then

- t e , e2 t cos 3 1, e2 t s in 31 ,

so any solut ion is of the form z( t ) = c l e t + c 2 e

- t + c3 e2 t cos 31 + c4 e2 t sin 31 .

(b) For the equation ji + ey = 0, k a real constant,

the roots of its characteristic polynomia l pl + k l are p = ± ik . Therefore a fundamental system of solutions i s

s in kl , cos kl ,


and any solution can be written as y(t) = a sin kt + b cos kt, a, b constant.

If A = jT+bi and r:t. = tan - 1 (bla), this can be conveniently written as yet) = A sin (kt + r:t.) ,

showing the amplitude A and in itial phase angle r:t. of the periodic solution. We now consider the case where the characteristic polynomial

L(p) of ( l ) has multiple roots. Recall that if L(P) = (p - 2Yg(P) and g(2) =1= 0, then 2 i s said to be a root with multiplicity r of L(p), where r is a positive integer. The fol lowing theorem gives a complete description of the fundamental system of solutions of ( \ ) .

T H E O R E M 3 . \ .6 . Let the distinct roots of the characteristic polynomial L(p) of( \ ) be denoted by }"j with respective multiplicities Pj , j = l , . . . , k . Then the functions t re).jt, r = O, l , . . . , pj - l , j = 1 , . . . , k , are a fundamental system of solutions of ( 1 ) .

Proof Evidently the eljt for j = \ , . . . , k are solutions by the corollary to Theorem 3 . 1 . 1 . From the properties of polynomials we have

k ( i) L Pj = n, and

j = 1 ( i i) if Pj > 1 , then

L(2) = L(A) = . . . = IJl'r 1 )(2) = 0. Consider the funct ion f(s, t) = eH ; it has continuous partial derivatives of all orders and therefore, writing ameS'lotm = (est) (m ) , we have

am ar = -� - -- f(s, t ) = ( t'es/)(m)

otm asr

for any positive integers r and m.

3. 1 The nth-Order Linear Equation 49

We use this fact to show the functions t re).jt are solutions as follows . Consider the function

<p(s, t) = (est)(II) + al (est)(n - l ) + . . . + an e" = L(D)est = estL(s) .

By the previous remarks we have

= �'[I:(s) + tL(s)] .

If s = Aj with Jlj > 1 , then the right side i s zero, which implies that te).jt is a solution of ( I ) . After r differentiations we would have the expreSSIOn

ar as� = ( t"est)(n ) + a l ( trest)(n - I ) + . . . + anC t"est)

= eS'[15r)(s) + rt 15r - I )(s) + . . . + trL(s)] ,

and, if Jl j > r , letting s = A j implies that tr e).jt is a solution of ( l ) . I t remains to prove l inear independence ; we indicate a proof

similar to that given in Theorem 3 . 1 .4. Suppose that some nontrivial l inear combination of the given functions were identically zero on - 00 < t < 00. This would be equivalent to the existence of polynomials pit), not all zero and of degree rj :s; Jlj - l , j = 1 , . . . , k such that

k L p/t)e).jt = 0, j = 1

- oo < t < oo .

Assume that P l (t) i= 0, then multiply the above sum by e - ). , t, and differentiate rl times-this annihi lates PI(t) . We are left with an expression of the form

k L q/t)e().r ).dt = 0,

j = 2 - 00 < t < 00 ,

where qP) are polynomials of degree rj • If we proceed as before we eventually arrive at an expression of the form

- 00 < t < 00 ,

where pet) i s a nonzero polynomial and r < s s k. This i s a contradiction, s ince a polynomial has only discrete zeros, and we conclude that the functions

form a fundamental system of solutions of ( I ) . Finally, we state a result analogous to that of Theorem 3 . 1 . 5 for

the case where the coefficients of the characteristic polynomial are real . The proof is a slight modification of the previous results and is left to the reader.

T H E O R E M 3 . 1 .7 . Suppose that the characteristic polynomial L(p) of( 1 ) has real coefficients, distinct real roots Aj with respective multiplicities Ilj ' j = 1, . . . , k, and distinct pairs of conjugate complex roots A j = u j + iv j ' A j + 1 = U j - iv j with respective multiplicities Ilj ' j = m, . . . , s. Then the real-valued functions

j = m, . . . , s ,

r = O, l , . . · , llj - l , are a fundamental system of solutions of ( 1 ).

Examples

(a) The characteristic polynomial of Z(6 ) - 8Z( 5 ) + 25z(4 ) - 32z(3 ) - i + 40i - 25z = 0

I S

L(p) = (p2 _ 1 )(p2 - 4p + 5)2 ,

and its roots are p = ± I , and p = 2 ± i with multipl icity 2 . Therefore a fundamental system of solutions i s

- I e , e2 1 cos t, e2 1 sin t, (b) Consider the equation

Z( 5 ) + 3Z(4 ) + 3Z( 3 ) + i = O.

te2 t cos t, te2 t sin t.

3 .2 The Nonhomogeneous nth-Order Li near Equation

Its characteristic polynomial i s

L(p) = p2(p + 1 ) 3 ,

and hence a fundamental system o f solutions i s

a n d any solution can b e expressed as z(t) = C1 + C2 t + (C3 + C4 t + Cs t2)e- t,

where C1 , • • • , Cs are constant.


We continue the discussion of the nth-order l inear equation

If b(t) is continuous on rl < t < r2 , and given initial conditions

z(t o) = Z o , z(t o) = z o , . . . , z(n - I )(t o) = z� - I ) ,

5 1

(6)

(7) where rl < t o < r2 , then by our previous discussion we know that a unique solution z(t) exists satisfying (7) and defined on rl < t < r2 •

Applying the method of variation of parameters we obtain the following result, corresponding to Theorem 2.6. 1 .

T H E O R E M 3 .2 . 1 . The solution 0/(6) satisfying initial conditions (7) is given by

Z( t) = cp(t) + W(O) - I f zi t) f eQ ' Sb(s) Wj(s) ds, j = 1 to

where

(i) cp(t) is the solution satisfying initial conditions (7) of the corresponding homogeneous equation,

(ii) Z 1 ( t ) , . . . , zn( t ) are a jill1damental system of solutions of the homogeneous equation and Wet) is their Wronskian, and

(iii) WP) is the determinant obtained from Wet) by replacing the jth column by (0, . . . , 0, 1 ) .


Therefore nothing essentially d ifferent is obtained from the case of constant coefficients. The only advantage is that we can explicitly determine a fundamental system of solutions for the homogeneous equation, and therefore determine Wit) .

Example : Consider the equation Z(3 ) - 3£ + 2i = log t, t > o. A fundamental system of solutions of the corresponding homogeneous equation is

and their Wronskian is

Computation gives

and, if to = I , the solution i s given by [ I I z(t) = cp( t) + -t J log s ds - 2e' f e - s log s ds

I I

, = cp(t) + 1 - i t + i t log t - e' r e - s log s ds • 1

, + iez l J e - Zs log s ds ,

1

where

cp(t) = C 1 + Cz e' + C3 eZ I , C 1 , Cz , C3 constant . It should be noted that the two i ntegrals i n the above expression can only be expressed as infinite series, but for practical purposes (that i s, numerical calculations), the solution is determi ned explicitly.

3 .2 The Nonhomogeneous nth-Order Linear Equation 53

A special method exists for solving (6) when bet) i s of a special form-namely, when i t i s a solution of a l inear equation with constant coefficients. This implies that there exists a polynomial operator

bi constant,

such that M(D)b(t) = O. By Theorem 3 . 1 .7 it fol lows that bet) must be expressible i n the form

k bet) = L <p/t)eAjt,

j = 1

where <p/t) , j = l , . . . , k are polynomials, and Aj , j = l , . . . , k are the roots of the characteristic polynomial M(p).

The method for solving (6) in thi s case i s called the annihilator method or method of undetermined coefficients, which we will briefly describe. We are given the relations

L(D)z = bet), M(D)b(t) = 0, from which it fol lows that

N(D)z = (ML)(D)z = M(D)L(D)z = M(D)b(t) = O.

The left side is a l inear equation with constant coefficients, so we can find a fundamental system of solutions Z l (t) , . . . , z,(t), and the solution of N( D)z = 0 i s given by

, z(t ) = L Ci z;( t), c i arbitrary constants .

i = 1

The relation L( D)i( T) = b(t) (8)

wil l serve to determine by comparison a number of constants, say Ci = Ci for i = 11, . . . , r. The solution of (6) is then given by

, z( t) = <pet) + L Ci Z;(/),

i = h

where <p(t) i s the solution of the homogeneous equation corresponding to (6) sat isfy ing the init ia l condit ions .

The evaluation of the constants Ci using relation (8) is not so difficult if we recall that some of the Zj(t ) for i = I , . . . , r satisfy the relation L(D)Zi( t ) = O. This is simply because the roots of the characteristic polynomial L(p) are a subset of the roots of the characteristic polynomial N(p) = M(p)L(p) . The following examples illustrate the method .

Examples

(a) Consider the equation z - z = e2 t( t 2 + 1 ) .

W e have L(D) = D2 - 1 ,

and

satisfies

M(D)b(t) = 0,

where M(D) = (D - 2)3 . It follows that

N(D) = (D - 2)\D2 - 1 ),

and the solution of N(D)z = 0 is given by

z(t ) = (cl et + C2 e - t) + (c3 e2 t + C4 te2 t + cs t 2e2 t)

= <p(t) + pet) .

Therefore we must choose C3 , C4 , and Cs so that

L( D)z = L(D)<p + L(D)f3 = 0 + P - p = e2 t(t 2 + 1 ) .

This leads to the relation

e2 t [3cs t2 + (8cs + 3c4)t + (2cs + 4C4 + 3c3)] = e2 t(t 2 + 1 ) ,

which, by comparing coefficients, gives Cs = t ·

3 .3 The Behavior of Solutions

The solution is then given by

z(t) = cl ef + Cz e - f + eZ f(� � - ! t + t t Z) ,

where CI and Cz are constants. (b) In the previous example, if we let

b(t) = e - f( t + 1 ) ,

55

then M(D) = (D + 1 )2 and N(D) = (D + 1 ) 3(D - 1) , so the solution of N(D)z = 0 is given by

z(t) = (cl et + c2 e - t) + (C3 te- t + c4 t 2e - t)

= qJ(t) + f3(t) .

The relation

p - 13 = e- t(t + 1 )

gives the result C3 = - 3/4, C4 = - 1/4, and the solution is then

z(t) = cl et + C2 e - t - ie- t(3t + t 2) .

3.3 The Behavior of Solutions

The solutions of the linear equation

L(D)z = z(n) + alz(n - l ) + . . . + an z = 0 (9)

where at , . . . , an are constants, are defined on - 00 < t < 00 . In many problems and applications we are interested in the behavior of the solutions as t approaches infinity. This behavior is related to the nature of the roots of the characteristic polynomial L(p) of (9), and we will discuss this relation in this section. Further discussion wil l be given in Chapters 4 and 5 , in which stability of solutions is discussed .

T H E O R E M 3 .3 . 1 . If all the roots of the characteristic polynomial L(p) of (9) have negative real parts, then given any solution z(t ) of (9) there exist positive numbers a and M such that

Iz(t) 1 � M e -ot, t � O. Hence, lim Iz(t)1 = o.

t - co

Proof If A j = Uj + h'j ' j = I , . . ' ., m are the distinct roots of L(p) , then by hypothesis Uj < 0, so we can find a number a > ° such that Uj + a < ° for j = I , . . . , m. A solution corresponding to Aj is of the form Zi(t) = t reAjt, and therefore I Z i(t )eat l approaches zero as t approaches i nfinity.

This implies that there exists a constant Mi > ° such that I Zi(t)eat l � Mi for t � 0, or I Z i( t ) 1 � Mi e - at for t � 0. Any solution z(t) of (9) can be expressed as z( t ) = I,7= 1 C i Zi(t ) , where Z i( t ) i = I , . . . , n are a fundamental system of solutions, and the C i are constants . If we let Mo = maxi l e d and M = Mo I,7= 1 Mi , then for t � ° n n

I z( t) ! � I, l ed I Zi( t) ! � Mo I, ! z i( t) 1 i = I i = 1

< M ( � M.) e - at = Me- at - 0 � , , i = 1

which is the desi red result .

C O R O L L A R Y. If all roots of L(p) with multiplicity greater than one hal'e negatice real parts, and all roots with multiplicity one hal'e nonpositil'e real parts, then all solutions of (9) are bounded for t � 0.

Proof If Aj = Uj + il'j for j = I , . . . , I' are the roots with mult iplicity one, then uj � 0, and a solution zit ) corresponding to Aj satisfies

I z/t) 1 = {(eUjt cos Vjt)2 + (eUjt s in Vjt) 2 } 1 / 2 = eUjt � 1

for t � 0. The remain ing roots Aj , j = I' + I , . . . , n have multiplicity greater than one and negative real parts ; hence with the same notation as before we have

I z( t) 1 � Mo it! I Z i( t) 1 � Mol' + (=� l Mi) e - at

for t � 0, which implies that z(t) i s bounded . There is only one drawback to the above resu lts-we must know

all the roots of the characteristic polynomia l L(p) to determine the

3.3 The Behavior of Solutions 57

behavior of solutions of (9) . The following theorem gives implicitly a test for the vanishing of solutions as t approaches infinity based on the coefficients of (9). Its proof is not given.

T H E O R E M 3 . 3 .2 (Routh-Hurwitz criteria) . Given the equation (9) with ai ' i = 1 , . . . , n, real, let DI = a i ' and for k = 2 , . . . , n let

a 1 a 3 a s aZ k- 1 1 az a4 aZk - Z ° a , a 3 aZ k- 3

Dk = det ° 1 az . a Z k - 4 ,

° ° ° ak where aj = ° if j > n. Then the roots of L(p), the characteristic polynomial of (9) , have negative real parts if and only if Dk > ° for k = 1 , . . . , 11 .

The test becomes impractical for large n ; for n = 2, 3, and 4 the results are as fo l lows .

The roots of L(p) have negat ive real parts if and only if n = 2 : 11 = 3 : n = 4 :

a I and az are positive, a i ' az , and a3 are positive and a l a2 - a3 > 0, a I ' az , G 3 , and a4 are positive and G l az a3 - a� - a4 ai > o.

Final ly, for the roots of L(p) to have negative real parts the fol lowing necessary condit ion i s often useful . We assume that the al , · · . , an are rea l .

TH E O R E M 3 . 3 . 3 . If the roots of L(p) , the characteristic polynomial of (9 ) , hare negatice real parts, then the a I ' . . . , an are positive.

Proof The polynomial L(p) can be factored into terms of the type p + a and/or pZ + bp + c, a, b, and c real . Since the roots of L(p) have negative real parts, this implies that a > 0, b > 0, and c > 0, which impl ies that the coefficients of L(p) are positive.

D E F I N I T I O N : The characteristic polynomial L(p) of (9) i s said to be stable if al l its roots have negative real parts.

Examples

(a) L(p) = p3 + 3p2 + 2p + I IS stable, since its coefficients are positive and

a1a2 - a3 = 3 · 2 - 1 = 5 > O.

Therefore all solutions z(t) of

z(3 ) + 32' + 2t + z = 0

satisfy l im I z(t) 1 = O. t .... 00

(b) L(p) = p4 + 6p3 + 7p2 + P + 2 is not stable, since

a1a2 a3 - a� - a4 ai = 6 · 7 · 1 - 1 - 2 . 62 = - 3 1 < O.

(c) L(p) = p5 + 3p4 - 2p + 1 is not stable, since a4 = 2 - < O.

3.4 The First-Order Linear System

We will complete our discussion of the l inear equation with constant coefficients by considering the first-order system

or

n Xi = L a ij xj , j = 1

X = Ax,

i = 1 , . . . , n, ( 1 0)

where A = (ai) is an n x n matrix with real or complex constant coefficients, and x = x(t) = (x t ( t ) , . . . , xn(t» i s an unknown vector function. From our discussion in Section 2.2 it fol lows that given any initial condition

a unique solution x( t) exists satisfying the in itial condition . Furthermore, a fundamental system of solutions lP t (t) , . . . , lPn(t) of ( 1 0) exists. We will attempt to give a description of it.

By transposing ( 1 0) it may be written in the convenient form

L(D)x = 0,

3.4 The First-Order Linear System

where

L (D) =

a l l - D a 1 2 a l n a2 1 an - D a2n

59

Here D is the differential operator and L(D) i s a matrix operator. If I i s the identity matrix, then we may write

p 0 0 (a " a l n 0 p 0 0 . . . L (p) = : = A - pl.

an i ann 0 0 p

D E F I N I T I O N : The nth-order polynomial det L(p) = det(A - pI) is called the characteristic polynomial of the matrix A .

D E F I N I T I O N : A number A that i s a root of multiplicity m of det(A - pI) i s called a characteristic root of the matrix A and m i s its multiplicity. Example: If the matrix

then

- 1 2

- 1

- 1) - 1 ,

2

(4 - P A - pI = �

- 1 2 - p

- 1

- 1 ) - 1

2 - P and det(A - pI) = _ (p3 - 8p2 + 2 1p - 1 8) = - (p - 2)(p - 3)2 i s the characteristic polynomial of A. Therefore A = 2 i s a characteristic root of multiplicity 1 and A = 3 is one of multiplicity 2 . To show the relation between the characteristic roots of A and

the fundamental solutions of ( 1 0) we will need some prel iminary

results . If A is an n x n matrix and M(p) is any polynomial-for example,

M(p) = pm + a1pm - l + . . . + am ,

then we can construct the corresponding matrix polynomial M(A) = Am + alAm - 1 + . . . + am I,

where k t i mes

A k = A · A . . . . . . A .

If x(t) = (x l (t) , . . . , xit)), then M(A)x makes sense, and i f D i s the differential operator, then by M(D)x we mean the vector

M(D)x = (M(D)x l ' . . . , M(D)xn) .

L E M M A 3 .4. 1 . If x = xU) is a solution of ( 1 0), then so is X( k) , its kth derivative for any k. Furthermore, if M(p) is any polynomial, then M(D)x = M(A)x.

Proof Since x = x(t) i s a solution of ( 1 0), then x = Ax and therefore

d ( d ) dx dx -d

(Ax) = - A x + A - = 0 + A - = A(Ax) , t dt dt dt

which implies that y = Ax =. x is a solution of ( 1 0), and, furthermore, that (dfdt)(X) = X = A 2x. It follows by induction that

d _ X(k - 1 ) = X(k) = AkX dt

is also a solution for any k. From the relation X( k) = DkX = A kX and linearity we have M(D)x = M(A)x for any polynomial M(p) and any solution x = x(t) of ( 1 0) .

T H E O R E M 3 .4 . 1 . Suppose that 2 is a characteristic root of multiplicity m of A and Xo is a vector satisfying (A - 2l)mxo = O. If x = x(t ) is the solution of ( 1 0) satisfying x(to) = xo , then (A - 2l)mx(t ) == 0, - 00 < t < 00 .

3.4 The First-Order Linear System 6 1

Proof Since x = x(t) i s a solution, by the previous lemma we have

(i) (A - AJ)mx(t) = (D - A)mx(t), and (ii) y = y(t) = ( D - A)mx(t) i s a solution of ( 1 0) , since it is a l inear

combination of x and its derivatives. But by uniqueness y(to) = (A - AJ)mxo = 0 implies that yet) == 0 for - 00 < t < 00 .

C O R O L L A R Y. Under the previous hypotheses the solution x(t) oj ( 1 0) satisJying x(to) = Xo must be oJ the Jorm

- oo < t < oo ,

where the p i( t) Jor i = I , . . . , n are polynomials of degree s m - 1 .

Proof The relation ( D - A)mx(t) = ° for - 00 t < 00 i s equivalent to the n equations (D - A)mxi = 0, i = I, . . . , n, and by the results of Section 3 . 1 any solution must be of the form indicated .

To complete the description of the fundamental system of solutions of ( 1 0) we need the following result from l inear algebra.

If A I , . . . , Ak are the distinct characteristic roots of A with respective multiplicities m l , . . . , I11k so that L�=l m j = n, then to every Ai there corresponds vectors xij ,j = I , . . . , m i , such that

(i) (A - A ; IYx ij = 0, where r S m i ' and (ii) the collection of vectors xij ' i = I , . . . , k, j = I , . . . , mk is

linearly independent.

The proof of this result is beyond the algebraic prerequisite intended for this book.

Note that (i) implies that (A - A ; I)"" xij = 0, thus letting ..1. = A j , m = m j , and Xo = x( to) = xij ; then the above corollary implies that ( 10) has a solution

x = <p( t) = (P I ( t)e",i r, · · · , pit)e", i ') , <p(to) = xij ' ( 1 1 )

where the p,(t), s = I , . . . , n, are polynomials o f degree S ill i . Proceeding in this manner for every x ij ' we would obtain a set of 11

solutions of ( 1 0) , and these would be l inearly independent since their values at t = to are the l inearly independent set of vectors xij .

We thus obtain a fundamental system of solutions of ( 1 0) of the form ( 1 1 ) .

There is only one drawback : given ( 1 0), i t i s generally no easy task to find the vectors xi} described above. The following constructive method, however, obviates this difficulty. (a) Find the characteristic roots of A and their respective multi

plicities by computing the characteristic polynomial det(A - pl) . (b) If A. is a characteristic root of A with multiplicity m, first assume

a solution of ( 1 0) of the form <pet) = (al eAr, • • • , an eAt ).

Substitute this i n ( 1 0) , which will lead to a system of n equations in the n unknowns a i ' . . . , an '

Solve this system to determine any linearly independent solutions, and modify <pet) accordingly. (c) If m > 1 , next assume a solution of ( 1 0) of the form

<p(t) = «a1 + bl t )eAr, • • • , (an + bn t)eAt ),

and substitute i n ( 1 0) to obtain 2n equations i n the 2n unknowns ai ' . . . , an , bl , . . . , bn . Solve this system to determine any l inearly independent solutions and modify <pet) accordingly. Proceed in this manner, finally considering a solution of ( 1 0) of the form

<pet) = (P I (t )eJ. t, . . . , P n(t)eJ.t ),

where pJt) are polynomials of degree m - 1 with undetermined coefficients. Proceed as before . (d) Steps (b) and (c) will result i n m l inearly independent solutions

of ( 1 0) . Fol low these steps for al l characteristic roots to determine a fundamental system of solutions of ( 1 0) .

Examples

(a) Consider the system .Y = 5x + 3y, Y = - 3x - y,

and hence

3.4 The First-Order Linear System 63

and det(A - pI) = p2 - 4p + 4 = (p - 2)2 , SO A = 2 with multiplicity 2 is the only characteristic root . Assuming a solution of the form

lP l(t) = (ae2 t, be2 t)

and substituting leads to the relation b = - a, a arbitrary, so letting a = 1 , gives the solution

lP l (t) = (e2 t, - e2 t) .

Assuming a solution of the form

lP2(t) = « a + bt)e2 !, (c + dt)e2 t)

and substituting leads to the relations

b = 3a + 3c = - d, a, c arbitrary.

Letting a = 0, c = 1 gives the solution

lP2(t) = (3te2 t, ( 1 - 3t)e2 t) .

These are a fundamental system of solutions, since their Wronskian is ( e2t 3 te2! ) 4t W(t) = det _ e2! ( 1 - 3 t)e2 t = e ,

so any solution of the system is given by

lP(t) = (x(t), yet»� = Cl lP l (t) + C2 lP2(t),

Cl , C2 constant. (b) Consider the system

x = x - 3y,

and hence

A = G - 3) 1 '

y = 3x + y,

and det(A - pI) = p2 - 2p + 1 0, whose roots are A = 1 ± 3i. In this case we consider a solution of the form

lP(t) = (ae! sin 3t + be! cos 31, cet sin 3t + de! cos 3t)

and substituting in ( 1 0) gives the relation

c = b, d = - a, a, b arbitrary.

Letting a = 0, b = 1 we have

({Ji (t) = (el cos 3t, el sin 3t) ,

and letting a = 1 , b = 0 we obtain

({J2(t) = (el sin 3t, - el cos 3t ) .

These are a fundamental system of solutions.

The method of variation of parameters i s used to obtain the solution of the nonhomogeneous system

x = Ax + B(t), A = (ai), B(t) = (b i (t), . . . , bit» .

The representation depends on being able to obtain a fundamental matrix <D(t) of the corresponding homogeneous system, which we are now able to do. The form of the solution is given in Theorem 2.4.2 and will not be discussed here.

Finally, analogous to Theorem 3.3 . 1 , we have the following result describing the behavior of solutions of ( 1 0) as t approaches infinity.

T H E O R E M 3 .4.2. If the characteristic polynomial L(p) = det(A - pI) of( 1 0) is stable, then

l im " ({J(t) ! ! = °

for any solution ((J(t) of( I O) ·

Proof If ({J i( t) = «({J 1 i(t) , . . . , ({Jn i(t» , i = 1 , . . . , n, is a fundamental system of solutions of ( 1 0) , then ((Jj;(t) = pj;(t)eM, i, j = 1 , . . . , n, where Pji(t) is a polynomial and A is a root of L(p) . Since the real part of A is negative, it follows from Theorem 3 .3 . 1 that there exist positive constants M and a such that l ({Jj; (t ) 1 � M e - al, t > 0, i, j = 1 , . . . , n. This implies that

n I! ({Ji( t) I I = L !({Jji( t) 1 � nM e- a" t � 0,

j = i

Problems 65

and for any solution cP(t) = Ii= 1 Ci CP i(t) we then have n

I l cp( t) 1 1 :s; I I c; l l l cpi( t) 1 1 :s; K e- at , t � 0, i = 1

where K = nM Ii'= l l e i l , which gives the desired result .

Problems

1. Determine first if the solutions of the following equations approach zero as t approaches infinity or are bounded . Then describe the realvalued solutions.

(a) Z(4) + 5£ + 4z = o.

(b) £ + 4i + 3z = O. (c) Z(3 ) + 7Z + 1 6i + 1 2z = o. (d) Z(4) + 2Z( 3 ) + 10£ = O. (e) Z(3 ) + £ + 4i + 4z = 0 z(O) = i(O) = 1 , £(0) = 2. (f) Z(4) + £ = 0 z(O) = i(O) = 0, £(0) = Z(3 )(0) = 1 .

2. Using the method of variation of parameters, find the solutions of the following equations. (a) £ + z = sec t tan t.

(b) £ � 3i + 2z = sin e- '.

(c) £ � Z = t- l � 2r 3 . (d) £ + z = cot t.

3. Using the method of undetermined coefficients, find the solutions of the following equations. (a) £ � 3i + 2z = sin t.

(b) Z(3 ) � i = t2e' . (c) £ + 4i + 3z = t cos t + e'. (d) £ � z = 4 sinh t.

4. (a) Find the general solution of the equation

y + q2y = A sin wt, q :::=:: O,

and show that, if w = q, then solutions are oscillatory (have an infinite number of zeros on � 00 < t < (0) but become unbounded. This phenomenon is known as resonance.

(b) Show that the addition of a damping term 2kjJ for k > 0 on the left side of the previous equation assures that all solutions are bounded regardless of the value of w.

(c) In the case (b) suppose that w '* q and let

cp = tan- l 2kw/(q2 - w2), - TT/2 :::;: cp :::;: TT/2.

Express the solution in the form

yet) = yet) + A G sin (wt - cp), where yet) is a solution of the homogeneous equation. The angle cp is called the steady-state phase angle and G is a measure of the gain.

S. An equation of the form

a, constant,

is called an Euler equation.

(a) Show that the substitution t = eU reduces the equation to a nth

order l inear equation with constant coefficients. Use this fact to

describe a fundamental system of solutions of (*) . (b) For the case n = 2 find conditions on the constants al and a2 that guarantee that all solutions approach zero as t approaches

infinity or are bounded.

(c) Find the solutions of the following Euler equations.

(i) t2i - 4ft + 6z = O.

(ii) t2i - 3tt + 5z = O.

(iii) 13z( 3 ) + It - z = 0:

6. (a) Verify that given f(t) continuous on -k < t < k, then

t t l 'n - l Z(/) = fa dtl fa dI2 ' " fa f(t.) dl.

is the solution of the equation

z(·) = f(t), z(O) = teO) = . . . = z(· - 1 >(0) = O. (b) Use Dirichlet's formula,

( dx r f(x, y) dy = ( dy ( f(x, y) dx,

to show that

I t (t - U)· - l z(t) = 0 feu)

(n _ I) ! du,

and verify that the last expression is the solution for the case n = 2.

Problems 67

7. Find a fundamental system of solutions of the following first-order

systems.

(a) x = � x + 8y, j = x + y.

(b) x = x + y, j = �2x + 3y.

(c) x = 2x + y, j = �x + 4y.

(d) x = x � 2y � z,

j = �x + y + z, Z = X - z.

(e) x = x � y + z, j = x + y � z, i = � y + 2z.

8. Find a fundamental system of solutions of the corresponding homo

geneous system, then use the method of variation of parameters to

find solutions of the following systems.

(a) x = � 3x + 2y + e - t, j = �2x + y + l ,

(b) x = x � y + e 2 t ,

j = �4x + y + t.

C H A P T E R

4

4.1 Introduction

Autonomous Systems

and Phase Space

The next two chapters will be devoted to a discussion of how solutions of certain differential equations or systems of differential equations behave, and the notion of stability of solutions will be introduced. The emphasis i s not upon finding solutions but upon describing them, and this qualitative study is one of the major aspects of the modern theory of ordinary differential equations.

To begin with we will consider two-dimensional systems of the form

x = P(x, y), y = Q(x, y), ( 1 )

where x = x(t) and y = y(t) are unknown scalar functions, and P and Q together with their first partial derivatives are continuous i n some domain r of the xy-plane. Such systems are called autonomous inasmuch as P and Q do not depend on t. If z = (x, y), then ( I ) is of the form z = fez) = (P(x, y), Q(x, y», and the hypotheses guarantee existence and uniqueness of solutions by Theorem 1 . 3 . 1 .

Some reasons for discussing systems of the form ( 1 ) are

(i) a more complete theory exists than for higher-dimensional systems, and

4. 1 Introduction 69

(ii) the geometry of the plane and of plane curves is available to il luminate the discussion.

Furthermore, in many cases the analysis of the important second-order autonomous equation

x + g(x, x) = 0, x = x(t) a scalar function,

can be considerably extended by transforming it into the system

x = y, Y = -g(x, y),

which is of the form ( 1 ) . We begin by giving some simple properties of solutions of ( 1 ) ,

and introducing some terminology.

L E M M A 4. 1 . 1 . If x = x(t) , Y = y(t), rl < t < r2 , is a solution of ( 1 ), then for any real constant c the functions

x l (t) = x(t + c), YI ( t) = yet + c) are also solutions of ( 1 ) .

Proof By the chain rule for differentiation i t fol lows that X I = x(t + c), YI = yet + c) . Since x = P(x(t), yet»� , y = Q(x(t), yet»�, replacing t by t + c gives

X I = P(x(t + c), yet + c» = P(XI ' YI ),

YI = Q(x(t + c), yet + c» = Q(xl o YI) ,

which implies that XI and YI are solutions. They are evidently defined on rl - c < t < r2 - c.

Remark: The above property does not usually hold for nonautonomous systems ; for example, a solution of x = X, Y = tx is x(t) = et, yet) = tet - et, and yet + c) = (I + c)et + c =1= tx(t + c) unless c = O.

As t varies, a solution X = x(t), y = yet) of ( 1 ) describes parametrically a curve lying in r. This curve is called a trajectory (orbit, characteristic) of ( 1 ) .

L E M M A 4. 1 .2 . Through any point passes a t most one trajectory.

70 Autonomous Systems and Phase Space

Proof Let C1 : x = X 1 (t), Y = Yl (t), and C2 : x = X2 (t), Y = h(t) be distinct trajectories having a common point

Then t 1 =1= t 2 , since otherwise the uniqueness of solutions would be contradicted . By the previous lemma,

is a solution, and (x(t2), y(tz» = (xo , Yo) implies that x(t) and yet) must agree respectively with x2(t) and h(t) by uniqueness. This implies that C1 and C2 coincide.

Note carefully the distinction between solutions and trajectories of ( I ) : a trajectory is a curve in r that is represented parametrical ly by more than one solution. Thus x(t), y(t) and x(t + c), yet + c), c =1= 0 represent distinct solutions, but they represent the same curve parametrically.

Example : As (I, varies between 0 and 2n the functions

x(t) = s in(t + (1,), yet) = cos(t + (1,) , - 00 < t < 00 ,

represent a n infinite number o f distinct solutions o f the system x = y, y = - x. They represent the same trajectory, the circle C: x2 + y2 = 1 .

Suppose there exists a solution x(t) = Xo , y(t) = Yo , - 00 < t < 00 , of ( I ) , where Xo and Yo are constants. Clearly no trajectory can pass through the point (xo , Yo), s ince UnIqueness would be violated . Furthermore, we have

x = 0 = P(xo , Yo),

since x(t) and y(t) are solutions. Conversely, if there exists a point (xo , Yo) in r for which P(xo , Yo) = Q(xo , Yo) = 0, then certainly the functions x(t) = Xo , yet) = Yo , - 00 < t < 00 , are a solution of ( I ) .

D E F I N I T I O N : Any point (xo , Yo) i n r at which P and Q both vanish i s called a critical point of ( I ) . Any other point in r is called regular.

4. 1 Introduction 7 1

Other names for critical points are singular points, points of equilibrium , and equi l ibrium states, and they may be thought of as points where the motion described by ( 1 ) is in a state of rest. We call attention to the following kinematic picture .

Consider the field of vectors Vex, y) = (P(x, y), Q(x, y» with (x, y) in r. Then ( I ) describes the motion of a particle (x, y) whose velocity (x, ]i) is given by Vex, y) at every point in r. Trajectories are fixed paths along which the particle moves independent of its starting point, and critical points are points of equilibrium .

Viewed in this way we call r the phase space of the system ( I ) . D E F I N I T I O N : A critical point (xo , Yo) of ( I ) is called an isolated critical point if there exists a neighborhood of (xo , Yo) containing no other critical points. We now introduce the notion of stability of a critical point or,

equivalently, stability of the solution x(l) = Xo , y(t) = Yo , - 00 0 there exists [) > 0 such that

(i) every trajectory of ( I ) in the [)-neighborhood of (xo , Yo) for some I = I I is defined for t I � { < 00, and

(ii) if a trajectory satisfies (i) it remains in the a -neighborhood of (xo , Yo) for t > ( I ·

If in addition every trajectory C: x = x(t) , y = y(l) satisfying (i) and (ii) also satisfies (iii) lim x(t) = Xo and l im yet) = Yo ,

t- oo t - oo

then (xo , Yo) i s said to be asymptotically stable. Finally, an isolated critical point that is not stable is said to be unstable.

The definition of stabi lity roughly states that (xo , Yo) is stable if once a trajectory enters a small disc containing (xo , Yo) it remains within a slightly larger disc for all future time. The above definition

is sometimes called stability to the right ; a similar definition can be given for stability to the left when t approaches - 00 .

Example : The point (0, 0) i s the only critical point of the systems

(a) x = y,

y = - x,

(b) x = - x,

y = -y,

(c) X = x,

y = y.

In (a) the trajectories are a family of circles C: x2 + y2 = , 2 , ° < ,2 < 00 given by the solutions

x(t) = , sin(t + a), y(t) = , cos(t + a). Then (i) and (ii) are satisfied with ,2 < (j = e but (iii) i s not ; therefore (0, 0) is stable.

In (b) and (c) the trajectories are a family of straight l ines C : y = (Yo/xo)x as well as the lines x = 0, y = 0, given by the solutions

x(t) = Xo e± (t - to), ye t) = Yo e± (t - to\ not both Xo and Yo equal to zero. Here the negative sign is used for (b), the positive sign for (c) . For (b) we have (i), (ii), and (iii) satisfied ; hence (0, 0) is asymptotically stable. For (c) either x(t) or yet) or both become infinite as t approaches infinity ; hence (0, 0) is unstable.

The phase space of the systems would look like these diagrams, in which arrows denote the direction of increasing time.

y y y

-------:;JE---- x

(al (b) (e)

73

4.2 Linear Systems-Constant Coefficients

In this section we will consider the l inear system x = ax + by, y = ex + dy, (2)

where a, b, e and d are real constants. Therefore we may let r be the entire xy-plane, and so all solutions are uniquely defined on - CIJ < t < 00 . Hence we can discuss the behavior of trajectories in

the phase plane of (2) . Why discuss systems of the form (2) ? First of all, a complete

description of the phase plane can be given , since solutions of (2) can be determined explicitly. Second, many systems can be expressed in the form

x = ax + by + e l (X, y),

y = ex + dy + e2(X, y) .

If e l and e2 are sufficiently small in the neighborhood of a critical point, we would hope that the behavior of trajectories is locally like that of (2) . Thus we need to know about l inear systems.

The point (xo , Yo) = (0, 0) is a critical point of (2), and we will assume there are no other critical points. This i s equivalent to assuming that ad - be =1= 0; the case ad - be = 0 is left for the reader to discuss in Problem 2 . The characteristic polynomial associated with (2) i s

det(A - pI) = det (a � p d � p)

= p2 _ (a + d)p + (ad - be),

whose roots are given by

A I , A2 = t[(a + d) ± Jra - d)2 + 4b�] .

Then, from our d iscussion in Section 3 .4, solutions are of the form x( t) = !(t)e). ' r, yet) = g(t)e).,t,

where f and 9 are polynomials of degree ::::; 1 . Since we are only interested in the behavior of trajectories, we will only need to know the nature of the roots Ai .

To simplify the description of the behavior of trajectories near the critical point (0, 0) , i t will often be useful to perform a l inear transformation of the form

� = !Xx + {3y, 'Y/ = yx + by, !Xb - {3y oF 0.

The point (x, y) = (0, 0) is mapped into (�, 'Y/) = (0, 0) , and conversely. Furthermore, such a transformation will only result in a rotation and a magnification or shrinking of trajectories, but wil l not distort their essential behavior near (0, 0) .

Case I: A I ' A2 are real , distinct, and neither is zero :

(a - d)2 + 4bc > 0.

The transformation

� = cx + (A I - a)y, 'Y/ = cx + (A2 - a)y

transforms (2) into the system

For in.,tance, since ad - bc = A I A2 and Al + A2 = a + d, we have

� = c.X- + (A I - a)y = cax + cby + (A I - a)(cx + dy)

= A I CX + ( - A I A2)Y + A l dy = A I CX + A I (A. I - a)y = A I � '

and s imi larly for 'Y/ . Therefore, to s implify the discussion we may a s well consider

the system

where Al and A2 are real . The solutions are of the form

where C I and C2 are arbitrary rea l constants .

(a) A I ' A2 have the same sign : ad - bc > 0 ; (i) both roots are negative : a + d < 0.

(3)

If A l < .1..2 < 0, then, as t approaches infinity, (x, y) approaches (0, 0), and y/x, the slope of trajectories near the origin, becomes infinite. If C I = 0 we have the rectilinear trajectory

and similarly if C2 = O. In this case we say (0, 0) is a stable node and the phase plane of

(3) looks l ike the following diagram, in which the arrows denote the direction of increasing time. The diagram will be rotated n inety degrees if .1..2 < A l < 0.

y

----------�----------x

For the corresponding phase plane of (2), the only essential changes in the diagram could consist of a rotation, and possibly the rectilinear trajectories wil l no longer be perpendicular. Evidently (0, 0) i s asymptotically stable.

(ii ) Both roots are positive : a + d > 0.

Then, if 0 < A I < A2 ' the diagram is the same with the arrows reversed . In this case (0, 0) is an unstable node.

(b) A I ' A2 have d ifferent sign : ad - bc < O.

If A2 < 0 < A I , then the rectil inear trajectories are

which approaches (0, 0) as t approaches 00 , and

which becomes infinite. If CI > 0, Cz < 0, then (x, y) approaches ( 00 , 0) as t approaches 00 or (x, y) approaches (0, - 00) as t approaches - 00 . A similar analysis can be made for the other possible values of CI and Cz .

In this case we say that (0, 0) i s a saddle point and is obviously unstable . The phase plane of the system (2) will resemble that below except for possibly a rotation and change of direction of the rectilinear trajectories .

Examples

For the systems

(a) x = - 3x + y, y = 4x - 2y,

(b) x = 2x + y,

y = x + 2y, (c) x = 2x + 3y,

y = x + y,

y

(a - d)Z + 4bc > 0 in all cases . The critical point (0, 0) is a stable node for (a), since ad - be > 0 and a + d < 0 ; whereas for (b) it is an unstable node, since ad - be > 0 and a + d > O. In (c) we have ad - be < 0, so (0, 0) is a saddle point. Case II: A I , Az are complex conjugate : (a - d)Z + 4be < O.

We may therefore assume that A l = U + iv and A2 = U - iv, where u, v are real numbers. The transformation

� = cx + (u - a)y, I] = vy,

transforms (2) into the system

� = u� - V17, � = v� + UI] .

Therefore we will consider the system x = ux - vy, y = vx + uy,

where U and v are real . (a) A I , A2 are imaginary : a + d = O.

Then Al = iv, A2 = - iv, U = 0 and (4) becomes

x = - vy, y = vx,

whose solutions are x(t) = ci cos(vt + a), yet) = ci sin(vt + a),

and the trajectories are a family of circles C = x2 + y2 = ci .

(4)

In this case (0, 0) is called a center and is stable but not asymptotically stable. The corresponding trajectories for the system (2) will be a family of ellipses. Note in (2) that if y = 0, then y = CX, which indicates that the direction of increasing time is clockwise if c < 0 and counterclockwise if c > o.

y

--�--r-+-�1--r-+---X

(b) A.1 , A.2 are complex : a + d =1= 0. Then solutions of (4) are

x(t) = cleut cos(vt + 0:), and the trajectories are a family of spirals

c : x2 + y2 = cie2ut•

The critical point (0, 0) is called a spiral point or focus and is asymptotically stable if a + d = u < 0, and unstable if u > 0. As before, the direction of increasing time is determined by the sign of c . Trajectories have no limiting direction, since y/z = tan(vt + 0:) has no l imit as t becomes infinite.

Examples

F or the systems

(a) x = - x + 3y, y = - 2x + y,

(b) x = x - 2y,

Y = 3x - 3y, (c) x = 2x + 2y,

y = - x + 3y,

--�'��------ x

we have (a - d)2 + 4bc < 0. The critical point (0, 0) i s a center (a + d = 0), a stable spiral point (a + d < 0), and an unstable

spiral point (a + d > 0), respectively. For (a) and (e) the direction of increasing time is clockwise (e < 0), whereas for (b) it is counterclockwise (e > 0) .

Case III:

a + d z A t = Az = -2 - :;t 0 ; (a - d) + 4be = O.

This is the case of a double root Al = A2 = u, where U = (a + d)(2 :;t 0 since ad - be :;t o.

(a) A special subcase arises when b = e = 0 in (2), which then becomes the system x = UX, Y = uy, whose solutions are of the form

The trajectories are then a family of straight lines C: y = (ez(el )x as well as the lines x = 0 and y = o.

Then (0, 0) is called a proper node and is asymptotically stable if a + d < 0, whereas it is unstable if a + d > o.

y

----�--��--------- x

(b) In the general case, we may assume that b :;t o . Then the transformation

a - d � = -- x + y,

2b

1 '1

= b x, transforms (2) into the system

. a + d i' � = -

2- '> '

(If b = 0, C 1= 0, then (2) is essentially in this form.) Therefore we may as well consider the system

x = ux, y = x + uy, ( 5) whose solutions are

x(t) = c jeut, y(t) = (C I t + c2)eut •

If Cj = 0, the rectilinear trajectory is the y-axis. Otherwise all trajectories are asymptotic to the y-axis , since y/x becomes infinite as t approaches infinity.

In this case the critical point (0, 0) is called a node or improper node ; it is asymptotically stable if a + d < 0 and unstable if a + d > O. The phase plane of (5) is sketched below and that of (2) will differ only by a rotation .

y

---------T.._------- x

Note : The cases /(a) and II/(b) are sometimes grouped under the heading improper nodes. The distinction i s that in the proper node trajectories approach or leave the origin in all directions, whereas for the improper nodes only one or two directions are possible.

Examples

The systems (a) x = - 2x,

y = - 2y, (b) x = 8x - y,

y = 4x + 4y, represent a stable proper node and an unstable node ( improper node) , respectively.

From the above analysi s we are led to the following result.

4.3 A General Discussion 8 1

T H E O R E M 4.2 . 1 . Given the system

x = ax + by, y = cx + dy, ad - bc #- 0,

where a, b, c, and d are real, then (0, 0) is an isolated critical point and is

(i) stable if the roots of the characteristic polynomial are purely imaginary,

(ii) asymptotically stable if the roots have negative real parts, or (iii) unstable if the roots have positive real parts.

4.3 A General Discussion

We return briefly to the general system x = P(x, y), y = Q(x, y), (6)

where P, Q, and their first partials are continuous in some domain r of the xy-plane, and we wi l l assume that r is maximal with respect to the last property. The domain r can be regarded as the phase space of the system , and hence r will contain trajectories of (6) and possibly critical points. To describe it further, we need the following theorem.

T H E O R E M 4 .3 . 1 . Let x = xU), y = yet) be a solution of (6) defined on a maximum interwl of existence rl < t < r2 · If X(t l ) = x(t2), y(t ] ) = y(t2), t ] #- t2 , then rl = - 00, r2 = 00, and the following two cases can occur:

(i) the solution is an equilibrium state of (6) and therefore x(t) = Xo , y(t) = Yo , -00 < t < 00, where (xo , Yo) is in r, or

(ii) the solution is periodic with period T > 0.

Proof By Lemma 4. 1 . 1 and uniqueness, we have x(t) = x(t + t] - t2), yet) = y(t + t l - t2),

and since (r I ' r 2) is maximal , this can only occur if r] = - 00 and r2 = 00. Let IT be the set of all periods of the given solutions, and IT is not empty since t l - t2 is a period . If C I and C2 are in IT, then c] ± C2 is in IT, and furthermore IT i s a closed set. For if cn , n = 1 , 2, . . . , are in IT and l im Cn = c, then by continuity of the

n -+ 00 solutions x(t) and y(t) it follows that c belongs to IT.

We assert that either IT is the set of all real numbers, and conclusion (i) of the theorem follows, or there exists a least positive number T in IT, and conclusion (ii) fohows. For if there were no least positive number in IT, then given any e > 0, there exists e in IT such that 0 < e < e. Given any real number r, there exists an integer m such that I r - mel � e < e, which implies r is in IT by closure, and therefore IT = R.

Now suppose T is the least positive number in IT. Then we assert that e in IT implies c = mT for some integer m. If not, there would exist some integer n such that Ie - nTI < T. But since Ie - nT I i s a period, this contradicts the minimality of T. Therefore IT i s the set of all integer multiples of T, and conclusion (ii) holds.

From the above theorem and the previous discussion in Section 4. 1 we can conclude that the phase space of (6) can only consist of (a) critical points,

(b) nonintersecting trajectories, or (c) closed curves called cycles, which are trajectories of periodic

solutions.

For the l inear systems discussed in Section 4.2 the phase space consisted of (a) and (b), or (a) and (c) i n the case of a center. For a nonlinear system all three may occur, as examples in the following section will show.

If C is any curve in the plane, then by a neighborhood of C we mean a set of points

{ (u, v) 1 1 1 (x, y) - (u, v) 1 1 < b, (x, y) belonging to C, b > O} .

Using this notion we can introduce the following important class of cycles.

D E F I N I T I O N : If K i s a cycle of the phase space of (6), then K is called a limit cycle if there exists a neighborhood of K such that any trajectory passing through the neighborhood is not a cycle. Example : The system x = y + x( l - x2 _ y2),

4.3 A General Discussion 83

has the periodic solution

x(t) = cos t, yet) = - sin t,

correspond ing to the cycle K: x2 + y2 = 1 . Other solutions may be obtained by using polar coordi

nates, and therefore

. xx + yy 1' = ---r

0 = xy - yx 1' 2

The system then becomes

and the solut ions are

OCt) = - (t - IX) .

These represent a fami ly of spirals that are inside K and tend

toward K as t approaches 00 when c > O. For c < 0 they are

outside K and tend towards K as t approaches infinity. Therefore

the cycle K is a l im i t cycle .

We may think of a l imit cycle K as a closed curve representing an isolated periodic solut ion of (6), and having the property that

trajectories near K spira l toward K, away from K, or both. In this

way we can define a stable, unstable or semistable l imit cycle.

Stable Unstable Semistable

The investigat ion of the existence of cycles or l imit cycles has led to

much fruitfu l research in this century, but its discussion i s beyond

the intent of this book.


4.4 Nonlinear Systems

We will now apply the previous analysis given for linear systems in Section 4.2 to systems of the form

x = P(x, y) = ax + by + e t (x, y),

y = Q(x, y) = ex + dy + e2 (x, y), where we assume that

(7)

(i) P, Q, and their first partials are continuous in some neighborhood of (0, 0),

(ii) ad - be #- 0, and

(iii) lim e j(x, y) = 0, i = 1 , 2, where r = Jx2 + y2 .

r --+ O r

This implies that (0, 0) is a critical point of (7), and given a system (7) satisfying (i), (ii) , and (iii) , we will say that (0, 0) is a simple critical point of (7) .

We wil l denote by V and V the respective vector fields defined by

Vex, y) = (P(x, y), Q(x, y», vex, y) = (ax + by, ex + dy),

for (x, y) near (0, 0) . In view of the assumption (iii) we might expect that the phase space of (7) near the origin would resemble that of the " l inearized " system

x = ax + by, y = cx + dy. (8)

The following theorem indicates that th i s i s the case. By I I I I we will mean the Euclidean norm .

T H E O R E M 4.4. 1 . The simple critical point (0, 0) of (7) is isolated and

lim I I � I I =

1 , r --+ O I I V I I l im (arg V - arg V) = 0 .

r --+ O

Proof V i s cont inuous and does not vanish on the circle I' = 1 , since (0, 0) i s the only critical point of (2) . If v = infr= 1 1 1 V I ! , then v > 0 and I ! V I I � vr for all r ; hence

l im I I � - � 1 1 � l im I I V - V i i = 0 r�O I I V I ! I I V I I r - O vr

by assumption (iii) . The last relation implies that V does not vanish near (0, 0), so the origin is an i solated critical point . The remaining statements fol low from the relations

and I II V I I I I I V

V I I I ! V I I - 1 � m - I I V I I '

tan - 1 U _ tan - 1 U = tan - 1 r u - v ] . I I + uu

We wi l l now describe the behavior of trajectories of (7) near (0, 0) , using the terminology of Section 4.2 . To do so we will use polar coordi nates. Suppose that C: x = x(t) , y = yet) is a trajectory of (7) ; then we may represent it as

c: I' = r(t) , w = wet) , r(t) > 0,

where x(t) = r( t ) cos wet) , y(t) = ret) sin wet) .

D E F I N I T I O N : Assume there i s a neighborhood U of the simple crit ical point (0, 0) of (7) in which

(i) all trajectories are defined on to < 1 < 00 or - 00 < 1 < 10 for some 10 ;

(ii) l i m r(t) = 0 or l im r(t) = O. t - - 00

Then (0, 0) is said to be (a) a spiral point i f l im I w(t) 1 = 00 or l im I w ( t ) 1 = 00 for all

trajectories in U, t -+ oo t - - oo

86 Autonomolls Systems and Phase Space

(b) a node if l im w(t) = C or l im w(t) = C, a constant, for al l t -> 00 t-- - 00

trajectories in U, or (c) a proper node i f it is a node, and for every constant C there i s a

trajectory satisfying l im wet) = C or l im wet) = C. t -> 00 t - - 00

D E F I N I T I O N : The simple critical point (0, 0) of (7) is said to be (a) a center i f there exists a neighborhood of (0, 0) containing

countably many closed trajectories, each contain ing (0, 0) and whose diameters tend to zero,

(b) a saddle point i f there are two trajectories approach ing (0, 0) along opposite d i rections, and all other trajectories close to e i ther of them and to (0, 0) tend away from them as t becomes infinite. We can now proceed to d iscuss how the trajectories of (7), with

the given assumptions, are related to the trajectories of (8) , the l inear system, near the simple critical point (0, 0) . First of all, if the trajectories of (8) satisfy

l im re t) = ° or l im re t) = 0, t -> co t - - 00

then so do the trajectories of (7) . Hence asymptotic stability or instability of the origin is preserved. This result wi l l be proved in Chapter 5 , when we discuss a general result for nonautonomous systems.

The fo llowing results are also true ; proofs are omitted s ince a detai led proof is req ui red for each i nd iv idua l case. (a) If (0, 0) is a spi ral point of ( 8) , i t i s a spi ral point of ( 7) . (b) I f (O, 0 ) i s a node of ( 8) , i t i s a node of (7) . (c) If (0, 0) is a saddle point of ( 8) , i t i s a sadd le point of (7) . (d) I f (0, 0) i s a proper node of (8) , i t i s not necessari ly a proper

node of (7) . However, if c ;(x, y), i = 1 , 2, are further restrictedfor example,

I . I c ; 1 Im --1. 1 + a r -> 0

is bounded for some rx > O-then (0, 0) is a proper node of (7) . (e) If (0, 0) is a center of (8), then it is either a center or a spiral

point of (7) . In the last case we may think of the elliptical characteristics of the linear system being sufficiently distorted by the s;(x, y) terms to make them spirals. An analysis of these terms or of dr/dt will often resolve the ambiguity.

Examples

(a) The motion of a simple pendulum is governed by the equation

e + 2k8 + q sin e = 0, k > 0, q > 0,

and by the substitution x = e, x = y this becomes the system

x = y, y = - 2ky - q sin x,

which we can rewrite as

x = y, y = -qx - 2ky + q(x - sin x).

Its critical points are ( ± nn, O), n = 0, 1 , 2, . . . , and the term sex, y) = q(x - sin x) satisfies the required assumptions near x = 0, so (0, 0) is a simple critical point.

We therefore consider the system

x = y, y = -qx - 2ky,

which has an isolated singularity at (0, 0) . If we assume, for example, that q > k2, then (0, 0) is a stable spiral point of the l inear system, and hence is a stable spiral point of the given system.

If we make the change of variable e = cp + n, we arrive at the equation

iP + 2k<iJ - qcp = 0, k > O, q > 0,

and a similar analysis shows that (0, 0) is a saddle point of the corresponding system. Therefore (n, 0) is a saddle point of the original system, and the phase plane of the pendulum equation conceptually might look like the following diagram.


y

x

(b) The system

x = y, y = 2x - X2 ,

has critical points at (0, 0) and (2, 0) . The first is a simple critical point (e(x, y) = x2) and is a saddle point. By making a change of variable x = Z + 2, we obtain the system

z = y, The point (0, 0) is a center for the corresponding l inear system ; this is the ambiguous case. By (e) above, a trajectory C passing through the positive x-axis near (0, 0) must intersect the negative x-axis . But the last system is unchanged if we replace y by - Y and t by - t, which implies that C is closed . Therefore (0, 0) is a center, so (2, 0) is a center for the original system.

x

Note that the phase plane of the last system contains all three ingred ients : critical points, nonintersecting trajectories, and cycles.

Problems

1. Describe the type and stabil ity of the critical point (0, 0) of the fol lowing linear systems.

Problems

(a) x = 3x + 4y,

Y = 2x + y.

(b) x = 3x,

y = 2x + y.

(c) x = x + 2y,

y = - 2x + 5y.

2. Given the linear system

x = ax + by, y = ex + dy,

(d) x = 3x - 2y,

Y = 4x - y.

(e) x = x + 3y,

y = - 6x + 5y.

(f) x = 3x + y,

y = -x + y.

a, b, e, d real,

89

where ad - be = 0, show that the phase plane of the system is one of the following : (a) a line of critical points with rectilinear trajectories approaching or going away from it, (b) a line of critical points with rectil inear trajectories parallel to it, or (c) every point is a critical point.

3. Consider the second-order equation for free oscillations,

x + 2kX + q2 x = 0,

and

k, q positive constants,

discuss its solutions with reference to the nature of the critical point (0, 0) of the corresponding linear system,

:K = y, y = __ q2x - 2ky.

4. Find the simple critical points of the following nonlinear systems and describe the local behavior and stabil ity of trajectories. Sketch the phase plane.

(a) x = - 4y + 2xy - 8, (d) x = -4x - 2y + 4,

y = 4y2 - x2 . y = xy.

(b) X = x2 - y2 - 1 , (e) x = -x2 - y2 + 1 ,

y = 2y. y = 2x.

(c) x = y - x2 + 2, (f) x = - x2 - y2 + 1 ,

Y = 2x2 - 2xy. y = 2xy.

S. Given the system

x = - x - y log - I r, Y = -y + x log- I r,

where r = vi x2 + -';;2 , show that (0, 0) is a simple critical point. Then

use the polar form to show that it is a spiral point, whereas it is a

proper node for the corresponding linear system.

6. Given the system

7T X = -y + xr2 sin - ,

r

7T Y = X + yr2 sin - , r

show that (0, 0) is the only simple critical point. Use the polar form

to show that

(a) the family of circles Cn : r = l in, n = 1, 2, . . . , are trajectories,

and

(b) the above family constitutes the only closed trajectories. Do this

by showing that trajectories between any two consecutive circles Co and Cn + I spiral away from or toward the origin, and trajectories

outside C1 become unbounded.

The example shows that, for a center, not all solutions near the origin

need be periodic.

7. Show that the origin is a spiral point of the system

x = -y - xr, y = x - yr,

whereas it is a center of the corresponding linear system.

8. Discuss the nature and stability of the simple critical points of the

systems corresponding to the following nonlinear equations. Sketch

the phase plane of the system.

(a) x + 6x - x2 + 4x = O.

(b) x + (X) 3 + X = O.

(c) X + HX) 2 + 2x2 - 2 = O.

(d) x + 3 1xl + 2x = O.

(e) x - x + x2 - 2x = O.

(f) x + ax + (3x3 = 0, a, (3 constants. Discuss each of the cases

a > 0 and (3 < 0, a < 0 and (3 > 0, and so on.

C H A P T E R

5 Stability for

Nonautonomous Equations

5.1 Introduction

We now extend our d iscussion of stability of solutions to a consideration of the general first-order equation

x = J(t, x) , ( 1) where x = x(t) = (x l (t) , . . . , Xn(t)) is an unknown n-dimensional vector function, and we assume that

J(t, x) = (fl ( t, x), . . . , fnCt , x))

is defined and continuous i n

r = {( t, x) I r 1 < t < 00 , I l x l l < a} .

Recall that i f x = (X l ' . . . , Xn), by the norm I l x l l we mean n

I l x l l = L I xJ i = 1

A solution (not necessarily unique) of ( I ) satisfying x(to) = Xo will be denoted by

x(t) = x(t ; to , xo) .

92 Stability for Nonautonomous Equations

D E F I N I T I O N : Let x(t) = x(t ; to , xo) be a solution of ( I ) satisfying

(i) x(/) i s defined on to � t < 00, and (ii) the point ( t, x(t» belongs to r for t � to

Then x(t) is said to be stable if

(a) there exists y > 0 such that every solution x(t ; to , XI ) satisfies (i) and (ii) whenever I l xl - xo l l < y, and

(b) given e > 0 there exists a b > 0, 0 < b � y, such that I I xo - xI I I < b implies

to � t < 00 .

A solution that i s not stable i s said to be unstable.

D E F I N I T I O N : The solution x(t) = x(t ; to , xo) of ( I ) is asymptotically stable if it is stable and in addition there exists p > 0, 0 < p � y, such that I l xo - xI I I < p implies

l im I l x(t ; to , xo) - x(t ; to , x l) 1 1 = o. ' .... 00

Geometrically, the definitions say that x(t) is stable if any other solution whose initial data is sufficiently close to that of x(t) remains in a " tube " enclosing x(t). If the diameter of the tube approaches zero as t approaches infinity, then x(t) i s asymptotically stable. Analogous definitions can be given for t approaching - 00 (stabil ity to the left) .

In Chapter 4 we considered the special case when the solution x(t ; to , xo) was identically Xo , corresponding to a critical point of the autonomous system.

Examples

(a) Every solution of the equation x = 0 i s stable, since - 00 < t < 00 ,

but n o solution i s asymptotically stable.

(b) Every solution of the equation x = - tx is asymptotically stable, since

I l x( t ; to , xo) - x(t ; to , x l ) 1 1 = I l xo - x I I I exp t(t� - t2) , - 00 < t < 00 .

(c) The scalar functions

x(t) = tanh ( t - to + k), - 00 < t < 00 ,

where k = tanh - 1 XO , - I < Xo < I , are solutions of the equation x = 1 - x 2 , and they are all asymptotically stable since they approach I as t approaches 00. The solution x = - 1 is not stable, whereas the solution x = 1 is asymptotically stable.

(d) The solution x = 0 of the equation x = x2 is unstable, since for to , Xo > 0 the solution

fai l s to exist at t = Xo I + to .

5.2 Stability for Linear Systems

The problem of stabi l ity of solutions of the l inear system

x = A( t)x (2)

will first be considered . Here x = x(t) = (x \ (t) , . . . , xn(t» is an unknown vector function, and the matrix A(t) = (aij(t» i s continuous for to � t < 00. Recall that the solution of (2) satisfying x(to) = Xo is then defined for f � to and given by x(t ; fo , xo) = (t)xo , where (t) i s the fundamental matrix satisfying (to) = [.

We wil l need the notion of the norm of a matrix.

D EF I N I T I O N : Given the 11 x 11 matrix A = (ai)' then I I A I I , the norm of A, is defined by

n I I A I I = I I O ij l .

i ,j = 1

94 Stability for Nonautonomolls Equations

Evidently I I II is a real valued nonnegative function defined on the set of n x n matrices, and if A = A(t) is continuous, then I I A (t) 1 1 is continuous. In addition it satisfies the properties

(i) I I A + B I I � I I A I I + l i B I I , I I AB I I � I I A I I I I B I I ,

(ii) I l eA 1 1 = 1 e i l i A I I for any scalar e, and (iii) I I Ax il � I I A l l l l x l 1 for any vector x,

as may be easily verified. In general, the notions of stabil ity of a solution and bounded ness

of a solution are i ndependent ; for example, the solutions x = t + Xo of x = I are stable but unbounded . However, in the case of l inear systems the two notions are equivalent by the following result .

T H E O R E M 5 .2 . 1 . All solutions of (2) are stable if and only if they are bounded.

Proof If all solutions of (2) are bounded, then there exists a constant M such that 1 I (t ) 1 1 < M, where (t) is the fundamental matrix of (2) satisfying <1>(10) = I. Given any 8 > 0, then I l xo - xI I I < 81M implies that

I l x(t ; to , xo) - x(t ; to , x l ) 1 1 = 1 I( t)(xo - x l ) 1 1 � M I l xo - x I I I < 8,

and hence all solutions are stable. Conversely, if all solutions are stable, then the solution

x(t ; to , 0) == ° i s stable ; therefore, given 8 > 0, there exists 6 > O such that I l xI 1 1 < 6 implies

1 1 0 - x(t ; to , x l ) 1 1 = II (t)x I I I < 8.

In particular, we can let X I be the vector with 6/2 in the ith place and zero elsewhere. Then

6 II (t)x I I I = I I ;( t) I I 2 < 8,

where <l>j(t) is the ith column of (t), and hence 1 I (t) I I < 2n86 - 1 = k. Therefore for any solution we have

I l x(t ; to , xo) 1 1 = I I (t)xo I I < k I l xo l l ,

and hence all solutions are bounded . We have previously considered in Section 3 .4 the l inear system

x = Ax, where A = (aij) is a constant matrix . Recall that the characteristic polynomial det(A - pI) of A is said to be stable if all its roots have negative real parts . We may now rephrase the result of Theorem 3 .4.2 as follows.

T H E O R E M 5.2.2. If the characteristic polynomial of A is stable, then every solution of x = Ax is asymptotically stable.

Proof If the characteristic polynomial is stable, then there exist positive constants R and r.x such that

1 1 $(t) 1 1 � Re - ar, t � to � 0,

where $(1) is the fundamental matrix satisfying $(to) = I. Since Re - at is a decreasing function, given e > 0, then I l xo - XI I I < eR - I eato implies I l x(t ; to , xo) - x(t ; to , x l ) 1 1 � 1 1$( t) l l l l xo - X I I I � Re - at I l xo - X I I I · The right side i s less than e for t � to and, furthermore, approaches zero as t approaches 00, so all solutions are asymptotically stable .

Furthermore, from our discussion of the nature of fundamental systems of solutions of the equation x = Ax, we can immediately obtain the fol lowing result.

T H E O R E M 5 .2 .3 . If the multiple roots of the characteristic polynomial of A have negative real parts, and the roots of multiplicity one have nonpositive real parts, then all solutions of x = Ax are bounded and hence stable.

A natural generalization is to consider systems of the form x = (A + C(t»x, (3)

where A = (ai) is a constant matrix , and the matrix C(t) = (cij(t» is continuous on to � t < 00. We might expect that if the characteristic polynomial of A were stable, then under suitable hypotheses on C(t) the solutions of (3) would be stable. This is the case, and to prove i t we will need the following lemma. It is often referred to as Gronwall 's inequality and is a most useful tool in the study of stability.

L E M M A 5 .2. 1 . If the nonnegative scalar functions u(t) and v(t) are continuous on to � t < 00 , a is a nonnegative constant, and

t u( t) � a + f v(s)u(s) ds

to for t z to , then

u(t) � a exp [{ v(s) dS] for t z to ·

Proof If a > 0, then the given inequal ity implies that

u( t)v( t) t � vet).

a + f v(s)u(s) ds to

Integrating both sides from to to t gives

IOg [ a + ( v(s)u(s) dS] - log a � { v(s) ds,

which impl ies that

u(t) � a + J' v(s)u(s) ds � a exp [J ' v(s) ciS] .

to to

If a = 0, the result holds for every a 1 > 0, and as a 1 approaches zero this implies that u(t) i s identical ly zero and the inequality is trivially satisfied.

Using the lemma we can now prove the fol lowing.

T H E O R E M 5 .2 .4. If the characteristic polynomial of A is slable, the matrix C(t) is continuous on 0 � I < 00, and

fOO I I C(t) 1 1 dt < 00 , o

then all solutions of x = (A + C (t» x are asymptotically stable.

Proof Using the expression given in Theorem 2.4.2 with B (t) = C (t)x(t), the solution x(t) = x(t ; 0, xo) of the equation must satisfy the expression

t x(t) = <1>(t)xo + f <1>(1 - s)C(s)x(s) ds . o

Here <1>( t) is the fundamental matrix of the equation x = Ax, and <1>(0) = I. Furthermore, the hypotheses imply there exists positive constants R and IX such that 1 1<1>(t) I I � Re- Ilt for t � 0, and therefore

t I l x( t) l l ellt :<:::; R I lxo l l + f R I I C(s) 1 1 I l x(s) 1 1 ellS ds. o We may now apply Lemma 5 .2 . 1 , which for t � ° gives

I l x(t) 1 1 ellt :<:::; R I lxo l l exp [ R f� II C(s) I I dS]

:<:::; R I lxo l l exp [ R (J II C(s) 1 1 dS] = M < 00 .

This implies that all solutions are bounded and hence are stable, and furthermore that they approach zero as t approaches 00 . Since the difference of any two solutions of a l inear system is also a solution, this implies that all solutions are asymptotically stable.

or

C O R O L L A R Y 1 . The conclusion of the theorem holds if the characteristic polynomial of A is stable, C(t) is continuous on ° � t < 00 , and I I C (t) 1 1 < c for t � ° with c sufficiently small.

Proof Proceeding as above we arrive at the i nequality

I l x(t) 1 1 ellt � R I l xo l l exp [ R f� I I c(s) 1 I dS]

:<:::; R I lxo l l eRe', t � 0,

I l x(t) l l :<:::; R I l xo l l e(Rc- ll)t, t � 0.

If c is small enough, so that Rc - IX < 0, the result follows.

C O R O L L A R Y 2. If all solutions of x = Ax are bounded, C (t )

is continuous on ° � t < 00 and f" I I C(t ) 1 1 dt < 00 , then all o solutions of x = (A + C (t» x are bounded and hence stable.

Proof If <1>(t) i s the fundamental matrix of the equation x = Ax such that <1>(0) = I, then by hypothesis there exists a constant K such that I I <1>(t) I I � K, ° � t < 00 . It follows that for x(t) = x(t ; 0, xo) we have

t I l x(t) 1 1 � K I l xo l l + f K I I C(s) l l l l x(s) 1 1 ds, t � 0, o and therefore

I l x( /) 1 1 � K I l xo l l exp [ K f� I I C(s) 1 1 dS]

� K I l xo l l exp [ K foGO I I C(s) 1 1 dS] = M < 00 .

Therefore all solutions are bounded, and hence stable by Theorem 5 .2. 1 .

Example : The second-order equations

(a) x + x = 0, 2x (b) x --- + x = 0, to > 0, t + to

correspond respectively to the systems (a') X = Ax, (b') X = (A + C( t» x , where

C(t) = (: �). t + to

and

A fundamental system of solutions of (a) is sin t, cos t so all solutions of (a) and (a') are bounded . A fundamental system of solutions of (b) is sin t - (t + to)cos t, cos t + (t + to)sin t, so a nontrivial solution of (b) and (b') is unbounded.

5.3 Two Results for Nonlinear Systems

For t � 0, we have 2 2

I I C( t) 1 1 = -- s - , t + to to

99

which can be made as small as desired by the choice of to . This example confirms that the hypothesis that II C (t) 1 I sufficiently small is not enough to insure that solutions of x = (A + C (t))x are bounded when those of x = Ax are bounded. Note that

t ( t + t ) 2 f l I C(s) 1 I ds = log __ 0 , a to

which becomes infinite as t approaches 00 .

5.3 Two Results for Nonlinear Systems

We will first consider the nonl inear equation x = A( t)x + J(t, x) , (4)

where A(t) = (auCt)) is a continuous n x n matrix defined on O s t < 00 , and the vector function

J(t , x) = (fl ( t , x), . . . , /,,( t, x))

satisfies

(i) J(t, x) i s continuous for I I x l l < a, 0 S t < 00, and (ii) lim I I J(t, x) l l / l l x l l = 0 uniformly with respect to t ; that IS,

I l x l l - o

I I J( t, x) 1 1 = o( l I x l l ) uniformly in t as I I x l l approaches zero .

Condition (i) assures local existence, but not necessarily uniqueness of solutions, and (ii) implies J(t, 0) = 0 ; hence x(t ) == 0, 0 S t < 00 , is a solution o f (4) .

In view of the condition (ii) , we might argue heuristically that since A(t)x + J(t, x) is very nearly like A(t)x for I I x l l near zero, then if solutions of x = A(t)x approach zero as t approaches 00, so do those of (4). This would be equivalent to asserting that the solution X(I) == 0 of (4) is asymptotically stable. Such an argument i s correct in the case A(t) = A, a real constant matrix , as the following theorem shows.

T H E O R E M 5 . 3 . 1 . If A is an n x n constant matrix v.,hose characteristic polynomial is stable, and J(t, x) satisfies conditions (i) and (ii) abore, then the solution X(/) == 0 of Ihe system

i = A x + J( t , x)

is asymptotically stable.

Proof We first show that the solution x( t ) = x(t ; 0, xo) is defined on 0 � I < 00 when Xo i s near zero. If <1>(1) is the fundamental matrix of the system i = Ax such that <1>(0) = 1, then by hypothesis there exist posit ive constants R and ('f. such that I I ( t) I I � Re - a t for t :2': O.

Since A i s a constant matrix, the solution x(t) must satisfy the relat ion

t x(t) = (t)xo + I (t - s)J(s , xes»� ds , o

which implies t I l x( t) 1 1 eat � R I l xo l l + I ReaS I I J(s, x(s» 1 1 ds. o

The first relation and hence the second is certainly val id for t in any interval [0, T) for which I l x( t) 1 1 < a if we assume I l xo l l < a.

From condition ( ii) i ffol lows that given any m > 0 there exists d > 0 such that for 1 :2': 0 and I l x l l < d we have I l f( t, x) 1 1 � m I l x l l . If we assume I l xo l l < d, then by continuity of x( t ) there exists I I > 0 such that I l xU) 1 1 < d for 0 � I < I I ' Therefore

. t I l x(t) 1 1 eat � R I l xo l l + I mReaS l l x(s) 1 1 ds • 0

for 0 � I < I I ' By Lemma 5 .2 . 1 this impl ies that

I l x( I) 1 1 � R I l xo l l e(mR - a)t ,

But Xo and m are at our d isposal, so we may choose m such that mR < ('f. and x(O) = Xo , so that I l xo l l < d/2R implies I l x( I ) 1 1 < dl2 for O � I < t l '

Since f( t, x) i s defined for l l x l l < a and 0 � I < 00 , this impl ies that we can extend the solution x( t) , which exists l ocal ly at every

5 . 3 Two Results for Nonlinear Systems 1 0 1

point ( t, x), t > 0, I l x l l < a, interval by interval, preserving the above bound . Hence, given any solution x(t) = x(t ; 0, xo) with I l xo l l < d/2R, i t is defined on 0 � t < 00 and satisfies Il x(t ; 0, xo) 1 I < d/2. But d can be made as small as desired, which implies that x(t) == 0 is stable, and mR < IX implies i t is asymptotically stable.

From this result immediately fol lows the statement made concerning the nonl inear autonomous systems discussed i n Section 4.4 : the asymptotic stabi l ity or instabi l i ty of trajectories of the l inear system is preserved .

In this case we considered the system x = ax + bY + l: t (x, Y), y = ex + dY + l:2 (X, Y), (5)

where ad - be =1= 0, eJX, y) were continuous together with their first partials and l im ei(X, y)/r = O.

r --> O Since the proof of Theorem 5 . 3 . 1 does not depend on the norm

chosen, i t fol lows that (a) i f the roots of the characteristic polynomial of A = (� �)

have negative real parts, then (0, 0) i s an asymptotically stable critical point of (5), or

(b) if the roots have positive real parts, then (0, 0) i s an unstable critical point of (5) .

The last statement means that trajectories near (0, 0) satisfy l im r( t ) = 0, and fol lows from the theorem by letting t approach

t -+ - oo - 00 i n the proof.

For the general system x = A(t)x + J(t, x) , (6)

with AU) a nonconstant matrix, the heuristic argument given at the beginning of this section fai ls . An example exists of a system for which 1( t, x) sati sfies conditions ( i) and ( ii) above, and for which the solutions of ,x = A (t)x are asymptotically stable, but the solution x(t) == 0 of (6) i s unstable .

One reason for th is deficiency is that the property of stabil ity is rather del icate and may not be mainta ined under smal l changes on the right s ide of (6) . A stronger definit ion of stability i s the following.

1 02 Stability for Nonautonomous Equations

D E F I N I T I O N : The solution x(t ; to , xo) = x(t) i s said to be uniformly stable if, given any e > 0, there exists D > 0 such that any solution xl (t) satisfying I l x(t l ) - x l (t I ) 1 1 < D for some t l � to exists and satisfies I l x(t) - xl (t) 1 ! < e for t � I I . Note the distinction between stability and uniform stability . In

the former a solution remains in an e-neighborhood of x(l ; to , xo) if it is close to the point Xo at time 10 ; other solutions may enter and leave the e-neighborhood at later times . In the case of uniform stability, once a solution enters the e-neighborhood of x(l ; to , xo) it remains there . Briefly stated, in the definition of stabil i ty the number D no longer depends on to .

Example : Consider the equation x = a(t)x, a(t) continuous on 0 =:::; t < 00 . Then

x(t ; t l , X I ) = Xl exp [{ a(s) ds l The solution X(I) == 0 i s uniformly stable if and only if the quantity

IX l l exp [f. a(s) dS] can be made uniformly small for sufficiently small value of Ix I I . Therefore x(t) == 0 is uniformly stable if and only if

expU: a(s) dS] is bounded above for t � t I � o.

The conclusion in the last example also fo l lows from the following result for l inear systems. We assume that A(t) i s continuous for t � to .

L E M M A 5 . 3 . 1 . All solutions of x = A (t)x are uniformly stable if and only if there exists a positive constant M such that

I I Cll( t)Cll - I (s) 1 1 < M, to =:::; s =:::; t < 00 , where Cll( t ) is any fundamental matrix.

5 .3 Two Results for Nonlinear Systems 1 03

Proof Given the solution x(t) = x(t ; to , xo) and any fundamental matrix (t), then for any tl � to we have the expression x(t) = (t) - l ( t I )X(t j ) . If x j (t) = (t) - l (t t )XI ( t j ) is any other solution, then given e > 0 the relation

implies

for t � t t � to , and hence x( t) is un iform ly stable . To prove the converse, note that the hypotheses imply that the

solution x( t ; to , 0) == 0 is un i form ly stable, then proceed as in the proof of Theorem 5 .2 . 1 .

Finally we wil l prove a result for the system (6), in which we assume that the matrix A(t ) i s continuous for t � to and f(t, x) satisfies

(i') f(t, x) i s cont inuous for I l x l l < a, t � to , and

(it) there exists a cont inuous nonnegative function !Yo(t) such that to !Yo ( t ) dt < 00 and fa

I l f( t , x) 1 1 � ct(t) I l x l l ·

Again note that (ii') impl ies that xU ) == 0, to � t < 00 , is a solution of (6).

T H E O R E M 5 . 3 .2 . Suppose that the solutions of x = A(t )x are

uniformly or uniformly and asymptotically stable, and f(t, x)

satisfies conditiol1s( i ' ) and ( it) above. Then the solution x( t ) == 0 of the system

,x = A( t)x + f(t, x) is Ulli{orm(r or uniformly and asymptotically stable .

Proof Since a l l solut ions of the l i near system are un i formly

stable, there exists by Lemma 5 . 3 . 1 a constant M such that for any

fundamental matrix we have

to � s � t < 00 .

If tl � to , then any solution x(t) for which I l x(tl ) 1 1 < a satisfies the relation

I x(t) = (t)- I(t l )X( t I ) + J (t)- I(S)!(S, xes)) ds,

I ,

for tl < t < T, where I l x(t) I I < a for t l � t < T. Therefore I

I l x( t) 1 1 � M I l x(t I ) 1 1 + M J et(s) I l x(s) 1 1 ds, I ,

and by Lemma 5 .2 . 1 this implies

I l x(t) 1 1 � M I l x( t I ) l l exp [ M f et(s) dS] � M I I X(t l ) l l expU� et(s) dS] = K I l x( t I ) I I .

But given any e > 0 with e < a, then I l x(t I ) I I < eK - 1/2 implies that I l x(t) 1 1 < e/2, and an argument similar to that given in Theorem 5 .3 . 1 implies that I l x(t) I I < e/2 for t � t 1 . Therefore the solution x(t) == 0 is uniformly stable.

Finally, if the solutions of the l inear system are in addition asymptotically stable, then l im I I (t) I I = 0; hence, given any Xo

such that I l xo II < a and any e > 0, there exists a To > to such that I I (t)xo I I < e for I � To . For the solution x(t) = X(/ ; to , xo) we then have for I � To

I l x( t) 1 1 � I I (t)xo I I + r 1 I (t)- I (S) 1 1 1 1!(s, x(s) 1 1 ds to

� e + r Met(s) I l x(s) 1 1 ds. to Again, by Lemma 5 .2 . 1 , this implies

I l x( t) 1 1 � e exp [ M f� et(s) dS] = eL,

and since e was arbitrary and L does not depend on e or To , we can conclude that lim I l x( t) 1 1 = O. Therefore the solution x(t) == 0 i s in

I .... 00 addition asymptotically stable .

105

5.4 Liapunov's Direct Method

We will now briefly d iscuss an important method of studying the stabil ity of solutions of the equation

x = f(t, x), (7) where x = x(t) = (x l ( t) , . . . , xit» i s an unknown vector function. The method i s known as Liapunov's direct or second method and depends on being able to construct a particular type of function V(t, x) from which the stabil ity or instability of the solution in question can be determined .

We assume that f(t, x) = (fl (t, x), . . . , /net, x») satisfies the fol lowing conditions :

(i) f(t, x) i s defined and continuous i n r = {( t , x) l l l x l l < a , /" 1 < t < oo } ,

(ii) a condition assuring uniqueness of solutions x(t ; to , xo) of (7) is satisfied at every point ( to , xo) in r, and

( iii) f(t, 0) = 0 for all t, and hence x(t ; to , 0) == 0 is a solution of (7) for to > rl •

For geometrical convenience by I l x l l we will mean the Euclidean norm of x. Some prel iminary definitions are now needed .

D E F I N I T I O N . The class K consists of all continuous, real-valued, strictly increasing functions <p(r), o :s; r :s; a, which vanish at r = O.

Let 0 r1 and suppose that Vet, x) i s a realval ued function, continuous together with its first partial derivatives in the set

B = {( t , x) I to :s; t < 00 , I I x l l :s; b} .

Furthermore, assume that V(t, 0) = 0 for t � to .

D E F I N I T I O N . The function Vet, x) is positive (or negative) definite if there exists a function <p i n the class K such that Ve t , x) � <p( i l x l l ) for al l ( I, x) i n B.

or Ve t, x) :s; - <p( l I x l l )

1 06 Stability for Nonautonomolls Equations

Examples

(a) The function Vex, y) = X4 + y4

i s positive definite, since vex, y) � 1r4, where r = );:2 -+ y2 . (b) The function

Vet, x, y) = t(x2 + y2) - 2xy cos t is positive definite for t � 2, since Vet, x, y) � r 2 .

D E F I N I T I O N. The function V(t, x) is said to be descrescent or to admit an infinitesimal upper bound if there exists h > 0 and a function tjJ in the class K such that I Vet, x) 1 s tjJ( l l x l l )

for I I x l l < h and t � to .

Example: In the previous example both functions are descrescent with tjJ(r) = r4 and tjJ(r) = 3r2 , respectively .

Now, given any function Vet, x) as above and the equation (7), we denote by V' the function

n av av V' = V'et, x) = L -;- J;(t, x) + -. .

i = l UXi ot If x = x(t), to s t s t l , i s a solution of (7) , then we can consider that Vet, x(1 » = V(t) . In this case V' = Ii is the derivative of V along the solution x(t) , and for simplicity we shall say that V ' is the derivative of V.

T H E O R E M 5 .4 . 1 . If a function Vet, x) exists that is positive definite, and whose derivative V ' is nonpositive, then the solution x( t) == 0 of (7) is stable.

Proof Since V is positive definite, there exists a function cp in K such that 0 < cp( l ! x l l ) s Vet, x) for 0 0, let

me = min cp( l I x l l ) , I I x l l =,

5.4 Liapunov's Direct Method 107

so m, > O . Since V is continuous and Vet, 0) = 0, we can choose (j > 0 so that V ( to , xo) < m, if I l xo l l < (j . Furthermore, since V ' i s non positive, t l � to and I l xo l l < (j implies

V(t I , x(t I ; to , xo) � V(to , x(to ; to , xo))

= V( to , xo) < m , .

Now suppose that for some I I > to we have I l x( t l ; to , xo) 1 1 = 8 when I lxo l l < (j. This would imply that

V(t l ' X(t l ; to , xo)) � <p( l l x( t l ; 10 , xo) l l ) = <p(8) � m" which is a contradiction. Therefore, i f I l xo I I < (j, then the solution x(t ; to , xo) i s defined for t � to and sati sfies I l x(t ; to , xo) 1 1 < 8, which implies that the solution x(t ) == 0 is stable.

To ensure that the zero solution of (7) is asymptotically stable, stricter conditions on the function V and its derivative are required.

T H E O R E M 5.4.2. If a function V ( t, x) exists that is positive definite, descrescent, and whose derivative V ' is negative definite, then the solution x(t ) == 0 of (7) is asymptotically stable .

Proof By the previous result the solution x(t) == 0 i s stable ; therefore, given 8 > 0, suppose there exists (j > 0, ), > 0 and a solution X(I ; to , xo) of (7) such that

1 � 10 , I l xo l l < (j . Since V ' is negative definite, there exists a function ,), in K such that

V'( t, x( t ; to , xo» � - ')'( i l x(t ; to , xo) I I ) .

Furthermore, since I l x( t ; to , xo) I I � A > 0 for t � to , there exi sts a constant d > 0 such that

V'( t , x(t ; to , xo» � - d < 0, This implies that

Vet , x(t ; to , xo» = V(to , xo) + r V' dt to

� V(to , xo) - d(t - to),


and for sufficiently large t the right side wi l l become negative, which contradicts V being positive definite. Hence no such A exists and, s ince Vet, x(t ; to , xo)) is a positive decreasing function, it fol lows that l im Vet, x( t ; to , xo)) = O. Therefore l im I l x( t ; to , xo) 1 1 = 0 and

t - oo this implies that the solution x(t ) == 0 is asymptotically stable.

D E F I N I T I O N. A function Vet, x) satisfying the hypotheses of

Theorem 5.4. 1 is called a Liapunov function of the equation (7) . In the case of autonomous systems we may omit the dependence

of V on t, and therefore delete the fami ly K, which serves to ensure a uniformity with respect to t. Thus V = Vex) is a positive (negative) definite function if

( i) V is continuous together with i ts partials in some neighborhood of the origin , and

(li) Vex) � 0 (or s O) , with equality only when x = O.

For the case n = 2 the following geometric description i s helpful. For small positive c the curves Vex, y) = c constitute a family of concentric loops enclosing the origin . The hypotheses of Theorem 5 .4 . 1 imply that on small enough loops the direction of the vector field defined by the system x = P(x, y) y = Q(x, y) never points outward . Hence, once a trajectory of the equation i s trapped inside such a loop, it cannot escape. The hypotheses of Theorem 5 .4.2 imply that the vector field points inward .

Liapunov functions have been described for certain classes of differential equations, but how to proceed with any particular equation is partially a matter of experience and ingenuity. Fortunately Liapunov functions are often closely related to certain physical characteri stics of the system described by the d ifferential equation.

Examples

(a) Given the second-order equation x + q(x) = 0,

5 .4 Liapunov's Direct Method 109

where q is continuously differentiable, q(O) = 0 and xq(x) > 0, we consider the corresponding system

x = y, y = - q(x).

The hypotheses insure that (0, 0) i s the only critical point. The total energy of the system is given by

y2 y2 x Vex, y) = - + E(x) = - + J q(s) ds,

2 2 0

where E (x) i s the potential energy integral . The function V is continuously differentiable, yeO, 0) = 0,

and xq(x) > 0 implies Vex, y) > 0 for (x, y) i= (0, 0) and therefore V is positive definite. Also,

V' = Vx x + Vy y = q(x)y + y( - q(x» = 0, and hence V is a Liapunov function and therefore the critical point (0, 0) is stable.

(b) For the system • 3 x = - y - x , y = x _ y3 ,

we consider the function Vex, y) = x2 + y2 . Certain ly V i s

positive definite and, furthermore,

V' = 2x( - y - x3) + 2y(x _ y3) = _ 2(X4 + y4)

is negative defin ite . Therefore the isolated critical point (0, 0) is asymptot ically stable.

(c) Suppose we are given a set of functions {<p(x)} called controls, such that for any <p the system x = lex, <p(x»

has un ique solutions, and f(O, <p(0» = 0, so that the origin is a critical point. The object i s to choose controls so that the trajectories of the corresponding system return to the originthat is, to an equi l ibrium position .

A natural choice for a Liapunov function is Vex) = d(x), the distance from x to the origin . The function d(x) is pos itive

1 1 0 Stability for NOllautollomous Equations

definite, and if we wish to avoid having trajectories tend away from the origin we must choose <[>(x) so that d'(x) < 0 for x i= O. Finally, to ensure that the trajectories always return to a state of equ i l ibrium, we must have d'(O) = 0 ; that is , d'(x) must be negative definite. Finally, we state the following i nstabi l i ty theorem for the

autonomous system x = f(x), (8)

where f(O) = 0, and f is continuous together with i ts first partial derivatives in some neighborhood r of the origin . The functions Vex) wil l as before be assumed to be continuous together with their first partial derivatives near the origin , say in r, and V(O) = O.

T H E O R E M 5 .4. 3 . If there exists a function V such that V ' is positive definite and in every neighborhood of the origin there is a point Xo where V(xo) > 0, then the solution x(t) = 0 of (8) is unstable .

Proof Let R > 0 be sufficiently small, so that the ball

S(R) = {x I l l x l l :s; R } l ies in r . Let M = max Vex) and M i s finite since V i s continuous.

I l x l l ,; R Choose r > 0 so that 0 < r < R and by hypothesis there exists a point Xo such that 0 O. A long the trajectory C : x = x(t ; to , xo) , t :2: to , V ' i s positive, and therefore V(x( t ; to , xo» , I :2: to , is an i ncreasing function and V(x( to ; to , xo» > O. This implies that C cannot approach the origin . Furthermore, since V' i s positive definite, the previous statement implies that

inf V'(x( t ; to , xo» = m > 0, t � to

and therefore V(x(t ; to , xo» - V(xo) :2: met - to)

for t :2: to . But the right side of the previous i nequality can be made larger than M for t sufficiently large, which impl ies that C must leave the ball S(R) ; therefore the origin i s unstable .

5.5 Some Results for the Second-Order Linear Equation

Example : For the system

1 1 1

consider the function Vex, y) = x2 - y2 . Then V i s continuous together with its first partials, V(O, 0) = 0, and V has positive values in any neighborhood of the origin. Furthermore,

V' = 2x(3x + y2) + ( - 2y)( - 2y + x3)

= (6x2 + 4y2) + (2xy2 - 2yx3),

and if Ix l and Iy l are sufficiently small the sign of V' is determined by the first term in parenthesis. Finally, V '(O, 0) = 0, and hence V ' is positive definite near the origin ; therefore the isolated critical point (0, 0) is unstable.

5 .5 Some Results for the Second-Order Linear Equation

In the study of differential equations, as i n many other fields of mathematics, the study of specific equations can give much more information than is obtained from general theorems. This i s certainly true for the equation

y + a(t)y = 0, (9)

one of the most widely discussed equations in the mathematical l i terature. It should be noted that the general second-order l inear equation

z + p(t)z + q( t)z = °

can be reduced to the form (9) by the transformation

z = y exp ( - 1 s: pes) dS) . Here y = y(t) is a scalar function, and we assume that aCt) i s

real-valued and continuous on ° � t < 00. We will give a few results, which compare the behavior of solutions of (9) as t approaches 00

with the solutions of a corresponding l inear equation with constant coefficients.

1 1 2 T H E O R E M 5 . 5 . 1 . If

rOt l a(t) 1 dt < 00 , I

Stability for Nonautonomous Equations

then l im yet) exists for any solution of (9) , and any nontrivial t-+ 00

solution is asymptotic to do t + dl for some constants do and d1 not both zero.

Proof Let y(t) be a solution of (9) satisfying y(to) = I , where 1 ::;; to < 00 . Integrating (9) twice from 10 to t and using Dirichlet's formula (see Problem 6, Chapter 3), we obtain the relation

t yet) = C I + t - f ( t - s)a(s)y(s) ds, to

where CI depends on to and Y(lo) . Therefore, s ince 1 � 1 , we obtain t l y(t) 1 ::;; ( i c i l + l)t + t f l a(s) l l y(s) 1 ds

to

or

l y�t) I ::;; ( l e l l + 1) + (s l a(s) I I Y�S) 1 ds for 1 � to . Applying Lemma 5 .2 . 1 we have

l y�t) I ::;; ( i c i l + l )eXP [(s la(s) 1 dS]

::;; ( I c l l + l )exp [J,�s l a(S) 1 dS] = c2 < 00 .

The constant C2 depends o n C1 , which i s a t our disposal, s o w e car choose to such that

1 - C2 fOOt l a( t) 1 dt > o. to

This implies that

r l a (s) l l y(s) 1 ds ::;; C 2 r s la(s) 1 ds < 1 � �

5 . 5 Some Results for the Second-Order Linear Equation

for t 2:: to . But t

yet) = 1 - J a(s)y(s) cis, to

1 1 3

and therefore lim y(t) exists and is not zero . Therefore yet) i s asymptotic to do t, do =f. 0, and now, using the result of Problem 6 , Chapter 2, we can conclude that

u(t) = ye t) {)y - 2(S) ds I

is another l inearly independent solution asymptotic to 1 / do. This com pletes the proof.

Example : The Euler equation

ji + mt- 2y = 0, m > i, has a fundamental system of solutions :

Y l ( t) = t l /2 cos(v log t) ,

yz ( t) = t l / 2 sin(v log t),

with v > 0. Therefore the assumption that l im aCt) = ° is not sufficient to guarantee that solutions of (9) behave l ike the solution of ji = 0. We wil l now consider the case where solutions of (9) behave like

those of the equation ji + y = O-that i s, are bounded and oscillatory. T H E O R E M 5 . 5 .2 If cp{t) is continuously differentiable,

l im cp(t) = 0, t -+ ex

and . 00 J 1 <,b( t) 1 dt < 00 , o then all solutions of the equation

ji + ( l + cp(t))y = ° are bounded.

Proof Multiply the equation by y, then integrate between 0 and t to obtain

(Y(t» 2 y2(t) t -

2 - + -

2- + Io <p(s)y(s)y(s) ds = c"

for any solution yet) where c, is a constant. Now integration by parts and transposition lead to the relation

t y2(t)[1 + <pet)] = 2c2 - (y( t» 2 + f <,b(S)y2(S) ds o

t s; 2c2 + f <,b(S)y2(S) ds, o

where C2 is a constant. Since lim <p et) = 0, we can choose to large 1-+ 00

enough so that 1 + <p( t ) ;;:: -!- for t ;;:: to , and therefore

l y2(t) 1 S; 4 1 c2 1 + 2 {1<,b(s) l ly2(S) 1 ds o for t ;;:: to ' Therefore by Lemma 5 .2 . 1 we have

l y2( t) 1 S; 4 1 c2 1 exp [2 {1<,b(S) 1 dS]

s; 4 1 c2 1 exp [ 2 ('I<,b(S) dS] = M < 00 ,

for t ;;:: to . Since yet) is continuous on 0 s; t s; to , it follows that yet) is bounded .

To show that solutions of the last equation are oscillatory we need the following result, which is a form of the Sturm Comparison Theorem. By an oscillatory solution we mean one having an infinite number of zeros on 0 s; t < 00 .

T H E O R E M 5 . 5 . 3 . If all nontrivial solutions of (9) are oscillatory and if b(t) is continuous and bet) ;;:: aCt ), to S; t < 00 , then all nontrivial solutions of

x + b(t)x = 0 are oscillatory.

5 . 5 Some Results for the Second-Order Linear Equation

Proof From the two equat ions we obtain the relation

yx - xji + [bet) - a( t)]xy = 0.

1 1 5

If t 1 and 12 are two consecutive zeros of a nontrivial solution y(t) of (9), we can assume that to � I I < 12 and that yet) 2: 0, I I � t � t2 • If X(/) is any solut ion of the second equation, then integrating the

previous expression from t I to 12 gives

12 x(t l )}'( t l ) - x(t2 )y( t2 ) + J [bet) - a(t)Jx( t)y( t) dt = 0.

I I

But if x( t ) d i d n o t change sign on I I � I � t2 , th is would lead t o a

contradict ion , since ji ( t l ) > 0, y( tz) < ° and y( t ) and b(t) - a( t ) are

posit ive on I I � I � I z . We may conclude that x( t ) has a zero in the

interval [t I , t 2 ] ' [ t fol lows that t he nontr iv ia l so lut ions 0f t he equat ion

Y + ( I + <p(t» y = 0, where l im <p(t ) = 0, are osci l latory. For to suffi-

cient ly large we h ave I + <p(t ) 2: 1 for t 2: 10 , and the so lut ions of

Y + 1Y = 0 are osc i l latory, so we can apply Theorem 5 . 5 . 3 . From the above proof we can actual ly i nfer more : if the two

solut ions xU) and y(t ) have a common zero at t = t I ' then the solu

tion x(t) of x + b(t )x = 0 must have a zero in the interval t l < t < t2 •

For, since x(t I ) = 0, we obtain the relation

. ( 2 x(tZ )y( t2 ) = I [be t) - a( t)Jx( t)y(t) dt .

" ( I

The right s ide is posit ive, so i f x( t ) were posit ive i n t I < t < t 2 we

would have a contrad ic t ion , s ince J" ( t2 ) < 0. We concl ude the d i scuss ion of the eq uation ( 9) by consideri ng

the case where a( t ) i s a str ict ly i ncreas ing unbounded funct ion and

we wi l l assume that aU) i s cont i n uously d i fferen t iable .

T H E O R E M 5 . 5 .4. rr l im a( t) = UJ monotonically, then a/l solu-

tions oj' (9 ) are bounded.

1 1 6 Stability for Nonautonomous Equations

Proof Multiplying the equation by y, integrating between 0 and t, and then integrating by parts lead to tbe relation

(y(t))2 + aCt) y2(t) _ II y2(S) des) ds = C I 2 2 0 2

for any solution y(t) of (9) where CI is a constant. We may assume that a(t) > 0 for t > 0 and therefore, since d(t) � 0, we have

I a(t)y2( t) I < J

ol I a(s)y2(s) I I des) I dS. 2 - I c l l +

0 2 a(s) Applying Lemma 5.2. 1 then gives

I a(t)y2( t) I (; > 0 for

I ..... 00 all t greater than some to , and we can conclude that the solutions of (9) are oscillatory. The amplitude of the osci llation never increases, as the following theorem shows. Note that between every two zeros of a solution will occur a zero of its derivative by Rolle's Theorem .

T H E O R E M 5 . 5 . 5 . Suppose that a( t) is continuously differentiable and a(t) > 0, d (t) � 0 on 0 � t < 00 . If yet) is a solution of (9) and tl and t2 are two consecutive zeros of its derivative, then ly(tz) 1 � ly(t I ) I .

Proof We may assume d(t) =t 0 , I I � t � /2 ; hence multiplying by 2y(t) and integrating from t I to t z gives

y2( t) + 2 I a( t)y(t)y(t) elt = O. ] 12 12

t l t l

Problems 1 1 7

By hypothesis the first expression is zero, and by a further integration by parts we have

12 a(t

Z)yZ( t

Z) - a( t l )yZ( t I ) = f d( t)yZ(t) dt.

I ,

Since y(t) does not change s ign in t l � t � tz , the solution y(t) is strictly monotonic in that interval. If yZ(tz) > yZ(t I)' then

{2d(t)yZ( t) dt < [a(tz) - a( t l )]y

z(t

z),

t ,

which implies that

a(t l )[yz(tz) - yZ( t I )] < O. This is a contradiction, and we conclude that yZ(tz) � yZ(t I ) ' which gives the desired result.

Problems

1. Given the nonhomogeneous system

x = A(t)x + B(t) , where A(t) and B(t) are continuous on to s: t s: 00 , prove that (i) if all solutions are bounded, then they are stable, and

(ii) if all solutions are stable and one is bounded, then all solutions are bounded.

2. Let eD(t) be a fundamental matrix of the system x = A(t)x, where A(t) is real-valued and continuous on 0 s: t < 00 . (a) Show that the transpose of eD- I (t) satisfies the matrix differential equation

X = -A(tfx, where A(t)T is the transpose of A(t). (b) Show that if the solutions of the system x = A(t)x are bounded and

lim inf J 'tr A(s) ds > - 00 ,

t - 00 0

then I leD- l (t) 1 1 is bounded. (Hint : Express eD- I (t) in terms of the adjoint matrix of eD(t) .)

3. Use the results of the previous problem to show that if the solutions of x = A(t)x are bounded, and

(i) the matrix C(t) is continuous on 0 ::;; t < 00 and r en I I C(t) I I dt < 00 • 0

and

(ii) lim inf C'tr A(s) ds > - CD , t -+ 'lJ .. 0

then the solutions of the system

x = [A(t) + C(t)]x

are bounded and hence stable.

4. The system x = A(t)x, where A(t) is real-valued and continuous on

o ::;; t < 00 , is said to be stable if all its solutions are stable. It is said

to be restrictively stable if it and its adjoint system x = -A(t? x are

stable. Prove the fol lowing.

(a) A necessary and sufficient condition for restrictive stability is

that there exist a constant M such that

t � O, s � O, where <l>(t) is the fundamental matrix satisfying W(O) = f. (b) If the system is stable and

lim inf etr A(s) ds > - 00 , r _ Xl ... 0

then it is restrictively stable.

(c) If the adjoint system is stable and

lim sup J 'tr A(s) ds < 00 ,

I _ CO 0

then the system is restrictively stable.

5. Let A(t) be a continuous real-valued square matrix on 0 ::;; t < 00 . Demonstrate the following.

(a) Every solution of the system x = A(t)x satisfies the relation

I lx(t) I I ::;; I lx(O) I I exp [D 'A(S) I I dSj .

Problems 1 1 9

(b) Use the previous result to show that if (' I IA(t) I I dt < 00 , then

every solution x(t) of the system has a finite limit as t approaches 00 .

(Hint : Show that A(t)x(t) i s integrable o n 0 ::;;: t < 00 .)

6. The system x = A(t)x is said to possess linear asymptotic equilibrium if, given any vector c, there exists a solution x(t) satisfying lim x(t) = c.

Using 5(b) above, show that the condition (' I IA(t) I I dt < 00 implies

that the system has linear asymptotic equilibrium.

7. Using appropriate Liapunov functions, determine the type of stability

or the instabil ity of the following systems.

(a) X = -x + y + xy,

(b) x = X - 3Y + X3,

y = -x + y - y2.

(c) X = -x _ 2y + xy2,

y = 3x - 3y + y3 . (d) x = -4y ± xe x + y

Y = 4x ± ye x + y

8. Let f(x) and g(x) be even and odd polynomials respectively and con

sider the second-order equation

x + f(x)x + g(x) = o.

(a) Show that the equation is equivalent to the system

x = y - F(x), y = -g(x),

where F(x) = f' f(s) ds.

(b) Let G(x) = f g(s) ds and suppose there exist positive constants

a and b such that g(x)F(x) > 0 for 0 ::;;: Ix l < a, and G(x) < b implies

Ix l < a. Show that this implies

y2 Vex, y) = "2 + G(x)

is a Liapunov function for the above system in the region Ix l < a, y2 < 2b, and that (0, 0) is an asymptotically stable critical point.

9. Apply the previous results to the van der Pol equation

x + e(xZ - 1 )x + x = 0,

to show that (0, 0) is an asymptotically stable critical point if e < O. Find the values of the constants a and b.

10. Consider the second-order l inear equation

y + p(t)y + q(t)y = 0,

where p(t) and q(t) are real-valued and continuous on rl < t < rz . Show that if Yl(t) and Yz(t) are a real-valued fundamental pair of

solutions, then Yl (t) must vanish between any two consecutive zeros

of Y2(t) (Sturm Separation Theorem).

1 1 . Estimate the number of zeros of any nontrivial solution of the

equation

Y + ky = O, k a positive constant,

that can be contained in the interval a ::;; t ::;; b. Use this result and

Theorem 5 .5 .3 to estimate the number of zeros of a nontrivial solution

of the given equation in the interval indicated.

(a) Y + 5ty = 0, 1 ::;; t ::;; 1 0.

(b) Y + t- zY = 0, 1 ::;; t ::;; 1 0.

12. Given the equation of Problem 1 0 above, show that a nontrivial

solution cannot have an infinite number of zeros on any closed

interval a ::;; t ::;; b contained in r l < t < r z .

Existence, Uniqueness,

and Related Topics

6.1 Proof of the Existence and

Uniqueness of Solutions

C H A P T E R

6

In Section 1 . 3 we stated without proof Theorem 1 .3 . 1 , a theorem giving conditions under which the existence and uniqueness of solutions of an ordinary differential equation are assured . The proof will be given in this sect ion, and depends on the method oj successive approximations.

This method, which is frequently used i n many different mathematical settings to prove existence of solutions, may be described as fol lows : we choose an in itial approximation xo(t) to a solution ,

based on the in it ial data. Furthermore, an algorithm i s devised whereby we can construct successive approximations Xl (t), x2(t), . . . , xn(t ) , . . . to a solution . Finally, we show that the sequence {(xnCt)} converges in some suitable topology to a solution x(t) .

Example : Given the d ifferential equation x = x, x(O) = 1 , where x = x( t) is a scalar function, then any solution must satisfy the relation

x(t) = 1 + {xes) ds. o

1 22 Existence, Uniqueness, and Related Topics

Therefore, given xn(1) , an approximation to a solution X(I), we

construct Xn + 1 (t) by the relation

t xn + 1 ( t) = 1 + f xis) ds.

o

As a first approxi mation, let xo( t ) == I , and we easily verify that

and, in general ,

the nth partial sum of the Taylor expansion of the solution

X(I) = e'.

x

X , (t) = I + t

o

To apply the method of successive approximations i n the proof of

Theorem 1 . 3 . 1 we wil l need some add it ional termi nology.

Given a posit ive i nteger n and any closed interval a s:: t :s; b, we denote by C[a, b] the set of a l l n-dimensional vector functions

x(t) = (x 1 (1) , . . . , xn( t » , which are continuous on a s:: t s:: b . We

introduce the fol lowing norm i n C[a, b] : n

I l x l l c = max I l x( t) 1 1 = max L I x/ t) I . a ,; t ,; b u ,; t ,; b j = 1

6. 1 Proof of the Existence and U niqueness of Sol utions 1 23

It is easily verified that C[a, b] becomes a metric space with the metric given by

d ist(x, y) = I l x - y l l e , where x = x( t) and y = y(t) are in C[a, b] .

Furthermore, note that given any Cauchy sequence {xs} in C[a, b ]-that is , a sequence of elements having the property that

l im I l x, - xs l l e = l im max I l x,( t) - x.( t) 1 1 = 0, r,S - 'XJ r, s - oo a :5 t ::; b

then, by the choice of norm, this i s equivalent to uniform convergence on a :-:::; t :-:::; b of the sequence {xs} . Since the l imit of a uniformly convergent sequence of cont inuous functions is a continuous funct ion, we may concl ude that C[a, b] i s a complete metric space.

Let the constant c > 0 and the point Xo in R" be given , and suppose f(t, x) is an n-dimensional vector-valued function that is continuous ' on the open set

E = r et , x) I a < t < b, ! I x - xo l l < c} . Let x = x(t ) belong to C[a, b] and suppose that the point (t, x(l » belongs to E for rl :-:::; t :-:::; r2 • Then, given any to such that rl < to < r2 ' the mapping

.six = y = yet ) = Xo + (1(s, x(s» lis,

• to

defines a continuous funct ion on rl :-:::; t :-:::; r2 • Since x = x(t) also belongs to C[r l , r2], we may concl ude that .sI i s a map from C[rl ' r2 ] i nto itse lf. S i nce y(lo) = xo , it fol lows that for a l l t in some neighborhood of to the point ( I, y(l» belongs to E.

Final ly , if we consider the d ifferen tial equation x = f(t, x), ( 1 )

then any solution x = xU) that sati sfies the in itial condit ion x( to) = Xo and is defined on r l :-:::; t :-:::; r2 also satisfies the relat ion

x(t) = Xo + ( ( s , x(s» ds , /" 1 :-:::; t :-:::; r2 · • to

124 Existence, Uniqueness, and Related Topics

But the right side of the last expression is .xl x, so we can conclude that any solution x = x(t), rl ::; t ::; r2 , of the differential equation ( 1 ) satisfies the relation

x = .xIx. (2)

Since x certainly belongs to C[rl ' r2 ], the last relation asserts that a solution x = x( t), r I ::; t ::; r 2 , of ( 1 ) is afixed point of the mapping .xl.

For the reader's benefit we restate Theorem 1 . 3 . 1 and then proceed to its proof.

T H E O R E M. Let the equation (*) x = f(t, x) be given, where f(t, x) is defined and continuous in some domain B contained in Rn + I and furthermore suppose that afl ax i , i = I , . . . , n are defined and continuous in B. Thenfor every point (to , xo) in B there exists a unique solution x = x(t) of (*) satisfying x(to) = Xo and defined in some neighborhood of (to , xo).

Proof The domain B i s an open connected set (assumed to be nonempty), and since (to , xo) belongs to B there exist positive numbers a and b such that the closed bounded set

r = {(t, x) l i t - to l ::; a, I l x - xo l l ::; b}

is contained in B. The functions f and aflaxi ' i = I , . . . . , n, are continuous in B, and therefore continuous in r. This implies that there exist positive numbers m and k such that

I If(t, x) 1 1 ::; m, j aJ;(t, X) j < k a - ,

Xj i , j = 1, . . . , n ,

whenever (I, x) i s in r. From the last inequality and the mean value theorem it follows that

I l f( t, X I ) -f(t, x2) 1 1 ::; nk I l x I - x2 1 1

whenever ( t , XI) and ( t, x2) belong to r, s ince r is convex. Now choose r > 0 such that

(i) r ::; a, (ii) r ::; blm, (iii) r < l ink, and let rr be the closed bounded subset of r defined by

rr = ret, x) l i t - to l ::; r, I l x - xo l l ::; b} .

6. 1 Proof of the Existence and Uniqueness of Solutions 1 25

The reasons for imposing the restrictions (i), (ii) , and (iii) on r will be clear shortly.

/ �� B --- / \ f 6 r, I " ··�' I I i -- � 2r - /

-------- ------------- 2a --------

Let rcr be the set of al l functions x(t) = (x1 (t), . . . , xit)) satisfying the following conditions :

(a) the function x(t) is continuous for I t - to l :::; r, and

(b) the point (t, x(t)) belongs to rr for I t - to l :::; r.

The conditions imply that I lx(t) - Xo I I :::; b for I t - to l :::; r, and therefore rcr is that subset of functions belonging to C[to - r, to + r], whose graphs l ie completely in rr .

For x = x(t) i n rcr let y = yet) be defined by

y = yet) = six = Xo + r I(s, xes)) ds, I t - to l :::; r. to

Then

I l y( t) - xo l l = I /(J(S , xes)) ds I I t :::; f I I I(s , x(s)) 1 1 ds :::; m i t - to l :::; mr :::; b,

to

and we see that condition (ii ) on the choice of r ensures that the graph of y = yet) = six is in rr for any x in rcr . Since yet) i s a continuous function, it follows that the choice of r ensures that s>l is a map from Cf,r into itself. Furthermore, note that y(to) = Xo , and hence y sati sfies the initial conditions.

Suppose that Xl = xl (t) and X2 = x2(t) belong to �r ' Then

I l dxI - dX2 1 1 = 1 1f: [/(s, x l (s» - f(s, xzCs» ] ds I I � f I I f(s, XI (S» - f(s, xis» 1 1 ds

to t

� nk f I l x I (S) - xis) 1 1 ds to

� nkr [ max I I x l ( t) - x2(t) l I ] = Ci I IX I - x2 11 c . I t - to l , 9

Thus condition (iii) on the choice of r ensures that Ci = nkr < 1 . It follows that

max I l dx I - dx2 11 = I l dx I - dx2 11 c I t - to l :5 r

for any Xl and X2 in �" where 0 < Ci < 1 . Since II I l c is a measure of the distance in �r ' the last relation implies that the distance between the images (under d) of two " points " in �r is less than the distance between them ; that is, d is contraction mapping from �r into itself.

We commence the process of successive approximations by letting xo(t) == xo , I t - to l ::; r, be the first approximation . Given no other information it is probably a poor but logical choice for a first approximation. We then define

and from our definition of the mapping d, x/to) = xo , and xj(t) belong to �r ' j = 0, I , . . . , k, . . . . Note also that

for any k, where by dk we mean the mapping d applied k times.

6. 1 Proof of the Existence and Uniqueness of Solutions 1 27

Let s and m be positive integers and s > m . Then by the contraction property, for the approximations Xs and Xm we have

I l xm - xs l l c = I l dxm - l - dXs- 1 1 l c � oe I l xm - 1 - xs- 1 1 l c

= oe I l dxm - z - dXs - z l l c � oezl l xm _ z - xs - z l l c � . . .

� oem I lxo - xs - m l l c •

By the triangle inequality we have

oem I l xo - xs-m l l c � oem

{ l i xo - x l l l c

+ I l x l - xz l l c + . . . + I l xs -m - l - xs-m l l } ,

and therefore

oem I l x o - xs - m l l c � oem I l xo - x i l i d l + oe + oeZ + . . . + oes-m - l }

Combining all the above, for any positive integers s and m , s > m, the successive approximations Xs and Xm satisfy

O < oe < 1 .

But the right side approaches zero as m approaches infinity, which implies that the sequence Xj = x/t), j = 0, 1 , 2, . . . , of successive approximations i s a Cauchy sequence.

Since C[to - r, to + r] is a complete metric space, it follows that there exists a continuous function x = x(t) defined on I t - to l � r such that

(a) lim xl t) = x(t ) uniformly on I t - to l � r, j-+ oo

(b) the point ( t, x(t» is contained in r, for I t - to l � r and x(to) = Xo .

Hence x = x(t) belongs to Cfl" and therefore for any j

I l dx - dxj ll c � oe I I x - xj l l c ·

1 28 Existence. Uniqueness. and Related Topics

But the right side approaches zero as j approaches co , which implies that

dx = l im dXj = l im Xj + l = X. j- oo j- co

Therefore x = x(t) is a fixed point of the mapping d and so is a solution of the differential equation satisfying x(to) = Xo . Existence of a solution is proved .

To prove uniqueness, suppose there exists another solution y = y(t) satisfying y(to) = Xo and defined on I t - to l $; r1 • Then there exists a positive number s $; min (r, r1 ) such that (t, yet)) l ies in r. for I t - to l $; s. Since x = x(t) and y = y( t) are solutions, we have

max I I x(t) - y(t) 1 I = I I x - Y l l c = I I dx - dy l l c I t - to I 5 '

o < rx < l .

This can hold only i f I I x - y l l c = 0 , which implies that x(t) = yet), I t - to l $; s; hence uniqueness i s proved, and this completes the proof of the theorem.

Example: Consider the equation

x(o) = 1 , and therefore J(t, x) = x2 + t2 , ( to , xo) = (0, 1 ) . Since B is the entire (t, x)-plane, let us choose as r the unit square

r = {Ct . x) I i t l $; 1 . I x - 1 1 $; I } .

Therefore a = b = I and

I �� I = 1 2x l $; 4

for ( t, x) in r, and hence m = 5 , k = 4. From the conditions (i), (ii). and ( iii) on r we deduce that ° < r $; t.

The mapping d is given by

yet) = dx = 1 + r [X2(S) + S2] ds, o I t I $; t.

6.1 Proof of the Existence and Uniqueness of Solutions 129

and, letting the first approximation be xo(t) = Xo = 1 , we have t

X l( t) = dxo = 1 + f (1 + S2) ds o

t3 = 1 + t + 3 ,

X2(t) = dX1 = 1 + I� [ (1 + s + s;r + S2] ds

2 1 2 1 = 1 + t + t2 + - t3 + - t4 + - t 5 + _ t7 3 6 1 5 63 '

and so forth . The successive approximations generated in this manner will converge uniformly to a solution x(t) satisfying x(O) = 1 and defined for It I ::; t. It should be noted that because of the restrictions on r, the

existence theorem is local in nature. The successive approximations converge to a solution satisfying the initial conditions and defined only in a neighborhood of to , even in the case where B is of the form

B = {( t , x) I - 00 < t < 00 , I I x - xo l l < b} ,

or B i s a l l of Rn + l . Whether the solution so obtained is actually defined for other t

is another question . In the example given at the beginning of this chapter the successive approximations converged to a solution defined for - 00 < t < 00. For the equation x = 1 + x2 the solutions are of the form x(t) = tan(t - h), so the successive approximations would converge to a solution defined in an interval of length less than 7r. In both cases B = R2 .

In the proof of the existence theorem the continuity in B of the partial derivatives 8fJ8xj was not explicitly used . As the proof indicates we could substitute the following Lipschitz condition .

There exists a constant k > 0 such that

I I J(t, X I ) - J(t , x2) 1 1 ::; k I l x l - x2 1 1

for every pair of points (t , X l ) and ( t , x2) in B.

This leads to the following result.


T H E O R E M 6. 1 .2 . Let the equation x = f(t, x) be given where f{t, x) is defined and continuous in some domain B contained in R" + 1 . If the above Lipschitz condition is satisfied, then corresponding to every point ( to , xo) in B there exists a unique solution x(t) of the equation satisfying x(to) = Xo and defined in some neighborhood of (10 , xo) .

The above theorem is actually a stronger result, since there exist continuous functions satisfying a Lipschitz condition but not having continuous first partial derivatives, for example, f{x) = I xl i n any neigborhood of x = O . However, in many cases i t i s easier t o verify the continuity o f the first partial derivatives than to determine whether a Lipschitz condition is satisfied . Furthermore, note that the Lipschitz condition need only be satisfied in some neighborhood of the point (to, xo) and the value of the constant k can vary with the choice of the domain B.

Finally, i t should be remarked that the existence theorem is an appl ication of a more general theorem, which states that a contraction mapping from a complete metric space into itself has a fixed point. However, although this principle is applicable in many different instances, it may turn out badly as an approximation technique. In the case of ordinary differential equations the convergence may be too slow, or the successive integrations may become cumbersome. If we are interested in finding the values of a solution near the in itial point x(1o), we should preferably use a more refined numerical technique.

6.2 Continuation of Solutions and

the Maximum Interval of Existence

In this section we will first discuss the question mentioned in the previous section-whether the solutions x = x(1) of the equation

x = f(t, x), (3)

which are uniquely defined in the neighborhood of an initial point, can be extended. We will also discuss the maximum interval of existence for l inear equations.

6.2 Cont in uat ion of Solutions and M aximum I nterval of Existence 1 3 1

Recall that given a point ( to , xo) i n the domain B, we were able to find a positive number r and a unique solution x = x(t) of (3) such that x(to) = xo , and x(t) was defined for I t - to l � r. Certainly the point ( to + r, x(to + r» i s in B (since B is an open connected set) and we can then repeat the method of successive approximations to obtain a solution defined in a neighborhood of the point { to + r, x{to + r».

This solution would then be defined on an interval to - r � t � to + r + 1"1., 1"1. > 0, and since it must agree with x(t ) for I t - to l � r, we are justified in calling it the continuation (to the right) of x(t) to the larger interval . Proceeding in this way we would eventually (since B is open) obtain an open interval rl < t < r2 and a solution x = x(t), defined on rl < t < r2 and such that x{to) = Xo .

This implies that the points (t , x{t » , r l < t < r2 ' are in B, and we will assume that r l and rz are finite . We would expect that if the points (r l , x(rl + 0» or (rz , x(rz - 0» were in the domain B, then under suitable conditions we should be able to extend the solution to a larger interval . By x(rl + 0) we mean the l imit of x(t) as t approaches rl from the right and x(rz - 0) is the l imit as t approaches rz from the left .

T H E O R E M 6 .2 . 1 . Letf(t, x) satisfy the hypotheses of Theorem 1 .3 . 1 in some domain B contained in Rn + I , and suppose x = x(t) , r l < t < rz , is a solution of(3) . Iff is bounded on B, then

l im x(t) and l im x(t) l -fo r t + 0

both exist, andfurthermore i[(rl , x(rl + 0» or (rz , x(rz - 0» are in B, the solution x(t) can be continued to the left or right.

Proof We may assume the solution satisfies x(to) = Xo for some ( to , xo) in B, rl < to < rz , and therefore x = x(t) satisfies

1 x(t) = Xo + f I(s, xes»� ds,

10

If bn = x(rz - l /n), then for n sufficiently large and m > n we have

r2 - I /m 1 1 1 1 I I bm - bn l l � f I I f(s, x(s» 1 1 dx � M - - - , r2 - I /n m n

where M is a bound for 1(/, x) in B. This implies that {bn} is a Cauchy sequence, from which it follows that l im x(t) exists, and

similarly for l im x(t) . t --+ r l + 0

Suppose that the point (r2 ' x(r2 - 0» is i n B ; then the function x(t) defined by

x(t) = x( t) , x(r2) = x(r2 - 0))

is a solution of (3), defined on r1 < t :-:; r2 • This follows from the relation

t X(/) = Xo + f I(s, xes)) ds,

to

which implies that the left-hand derivative x(r2 - 0) exists and equals l(r2 , x(r2)), which i s finite.

But B i s open, and by Theorem 1 . 3 . 1 there exists a solution <pet) of (3) passing through the point (r2 , x(r2 - 0)) and defined on some interval r2 - IX :-:; t :-:; r2 + IX, for some IX > O. Now define the function yet) as follows :

yet) = x(t),

yet) = <pet),

and we assert that y(t) i s a solution for r1 :-:; t :-:; r2 + IX . By uniqueness, x( t ) = <p(t) for r2 - IX :-:; t :-:; r2 (we may assume

that r1 :-:; r2 - IX) , so we need only show the existence and continuity of y(t) at t = r2 • But

yet) = x(r2) + r I(s, yes)) ds , r,

since <pet) is a solution, and furthermore

x(r2) = Xo + f'f(S, yes)) ds, to

since x(t) is a solution. This gives

yet) = Xo + f f(s, yes)) ds, to

/"2 :-:; t :-:; r2 + IX,

6.2 Continuation of Solutions and Maximum Interval of Existence 1 33

and since f and y(t) are continuous this implies that y(t) = f(t, y(t» , rl :::;; t :::;; rl + a, where one-sided derivatives are intended at the end points. Therefore y(t) i s a continuation (to the right) of x(t), and this completes the proof.

Example : Let f( t, x) = Xl and let B be the set

B = {(t , x) l - oo < t < 00 , I x l < oo } .

For the solution x(t) = - t - I , 0 < a < t < 00 , l im x(t ) exists t -+<% + O

and (a, x(a + 0» belongs t o B. Therefore the solution can be continued to the left, but it cannot be continued to the interval o :::;; t < 00 , since (0, x(o + 0» is not in B.

Given the solution x(t) of ( 3) satisfying x( to) = Xo , and whose graph lies in the domain B, let (m ] > m z) be its maximum interval of existence (see Section 1 .4) . By maximality the solution cannot be extended to the right or left of (m j , m l ) . ]f ml i s finite, we would expect that as t approaches m l from the left the values of x(t) become infinite or approach the boundary of B. The following theorem indicates that this is the case .

T H E O R E M 6.2 .2 . Let f( t, x) satisfy the hypotheses of Theorem 1 .3 . 1 in some domain B contained in Rn + l . Let x = x( t ) be a solution of(3) and let (m j , ml) be its maximum interval of existence. If ml is finite and E is any closed bounded set contained in B, then there exists an e > 0 such that the point (t, x(t» does not belong to E if t > ml - e. A similar statement holds ifm ! isfinite.

Proof Since E is closed and bounded and Rn + 1 - B = c(B) is closed , an elementary topological result gives

dist(E, c(B» = inf I I x - y I I = p > 0,

where the infimum is taken over all points x in E and all points y i n c(B) . Therefore if ( to , xo) i s in E, then the relation

I I (to , xo) - (t, x) 1 1 < p

implies that ( t, x) is in B.

Let E * be the closed bounded set of al l points whose distance from E is less than or equal to p12. Then E * contains E and i s contained in B , and there exist positive constants m and k such that

I I J( t , x) 1 1 � m, I I J(t, X l ) - J(t, x2) 1 1 � n k I l x l - x2 1 1

for a l l points ( t , x), ( t , xJ and (t, x2) in E * . Choose a > 0 and b > 0 such that a + b < p/2 . As in the proof

of Theorem 1 . 3 . 1 , choose r > 0 satisfying the relations

( i) r � a, ( i i) r � him, ( i i i) r < l in k .

Since the choice of m and k depended on E * , given any point (to , xo) in E, the solution x = <p(t) of ( 3) satisfying <p(to) = Xo will be defined for It - to l � r. We assert that the choice of e = r gives the desired result .

For suppose the given solution x(t) has the property that ( t l , X( t l » i s in E for some value of 1 1 > m2 - r. Then by uniqueness xU) must agree with the solution x = <p(t) , satisfying the initial conditions <p(t t ) = x(t t ) and defined for j t - l J i � r. Since m2 < 1 1 + r,

this contrad icts the assumption that (m I , 1112) is a maximal interval of existence for x(t) .

Examples

(a) Let B be the half-plane

B = re t , x) 1 0 < t < 00, x > O} .

Then the solut ion x( t ) = t t /2 of the equation .x = 2/x escapes

from every c losed bounded set in B because l im x( t) = O. In

th i s case li l t = 0 and 111 2 = 00 . (b) Let B be the ent i re L\'-plane. Then the solut ion xU) = tan ( t - h)

of the equat ion .x = 1 + x2 escapes from every c losed bounded

set because i t becomes unbounded . r n this case,

IT 1 l l 1 = - - + h 2

and IT

1 1 1 2 = - + h . 2 To concl ude th i s sect ion we w i l l g ive the proof of Theorem

2 . 1 . 1 , which we restate for the readers convenience.

6.2 Continuation of Solutions and Maximum Interval of Existence 1 35

T H E O R E M . Given the equation x = A(t)x + B(t), where A(t) = (aij(t» , i, j = 1 , . . . , n, and B(t) = (b l (t), . . . , bn(t» are continuous on rl < t < r2 , then for any initial value ( to , xo), rl < to < r2 , there exists a solution defined on r l < t < r2 and satisfying the initial value.

Proof We proceed as in the proof of existence and uniqueness of solutions. Therefore f(t, x) = A(t)x + B(t) and B is the set

B = {(t , x) I " 1 < t < r2 , - 00 < Xi < 00, i = 1 , . . . , n } .

Given ( to , xo) i n B and any continuous function x = x(t), r l < t < r2 , the mapping .xl is then defined by

y = y(t) = .xIx t

= Xo + J [A(s)x(s) + B(s)] ds, to

1" 1 < t < r2 •

Since A(t) and B(t) are continuous on rl < t < r2 , it fol lows that .xl is a map from e[1"1 , r2 ] into itself.

Let Sl and S2 be any real numbers such that r l < Sl < to < S2 < r2 , and let the initial approximation be xo(t) = XO , 1"1 < t < r2 • We will show that the successive approximations Xj + l (t) = .xIxj , j = 0, 1 , 2 , . . . , converge uniformly on S l ::::; t ::::; S2 ' Since S l and S2 are arbitrary, this will imply that the solution x(t ) = l im xP) i s

j� oo defined o n r l < t < r2 , which i s the desired result.

S i nce A( t ) and x l ( t ) are conti nuous on S I ::::; t ::::; S2 , there exist constants k > ° and c > ° such that

I I A ( t) 1 1 ::::; k , for S I ::::; t ::::; S2 . Therefore for S I ::::; t ::::; S2 we have

I I x 2 - x I I I = I I .([A(S)X \ (S ) - A(s)xo(s)] ds \ \ . 1

::::; , I I A(s) 1 1 I l x \ (s) - xo(s) 1 I lis ::::; kc It - t 0 1 , • 10

. 1 I I x 3 - x2 1 1 ::::; , I I A (s) 1 1 i l x2(S) - x l (s ) 1 1 ds

.. to


and in general

which implies that

But the last expression is the jth term of the series expansion for c exp[k(r2 - r] )] , and it approaches zero as j approaches infinity. This implies that the sequence xj(t) , j = I , 2, . . . , of successive approximations converges uniformly on s ] � t � S2 ' which completes the proof.

6.3 The Dependence of Solutions on Parameters

and Approximate Solutions

We first will consider the differential equation

x = l(t, x, A) , (4)

where t is a scalar variable, x = (x ] ' . . . , xn), and A = (A ] , . . . , Av) . Here 1 = (/] ' . . . , In) is an n-dimensional vector function defined on some region contained in Rn + v + ] . The vector A may be thought of as representing a set of parameters ; therefore, given fixed A = ..1.0 , we will denote a solution of (4) by x = x(t, ..10) '

If I satisfied the hypotheses guaranteeing existence and uniqueness of solutions for some domain B in Rn + v + ] , then we might expect that given a solution x = x(t, ).) of (4) satisfying x(to , A) = xo , then varying A sl ightly would only vary the solution slightly. This is equivalent to saying that solutions of (4) are continuous functions of the parameter A . The following theorem indicates that this is certainly true locally.

T H E O R E M 6 .3 . 1 . Let the equation (4) be giren where l( t, x, A) and

�flaXi are defined and continuous in some domain B contained in Rn + v + I . II (to , xo , ..10) belongs to B, then there exist positive numbers r and p such that

6.3 Dependence of Solutions on Parameters 1 37

(i) given any A such that I I A - Ao l l � p, there exists a unique solution x = X(t, A) of (4), defined for I t - to l � r and satisfying x(to , A) = xo ,

(ii) the solution x = x(t, A) is a continuous function of t and A .

Proof There exist positive numbers a, b, and p such that the closed bounded set

P = {(t, x, A) l l t - to l � a , I l x - xo l l � b , I I A - Ao l l � p}

belongs to B. Furthermore, there exist positive numbers m and k such that

I l f(t, x, A) I I � m,

i, j = I , . . . , n, whenever ( t, x, A) belongs to P. As before, choose r > 0 such that

(i) r � a, (ii) r � blm, (iii) r < l ink,

and let Pr be the set

Pr = {(t, x, A) l i t - to l � r, I l x - xo l l � b , I I A - Ao l l � p} .

Let �r be the set of all continuous functions x = x(t, A) having the property that the point (t , x(t, A» belongs to Pr for It - to l � r and I I A - Ao l l � p. This is equivalent to the relation I I x(t, A) - xo l l � b for I t - to l � r and I I A - ..10 1 1 � p.

Then for any function x = x(t, A) in �r let the map .91;. be defined by

.xI;.x = Xo + r 1(5, xes, A» ds , to

I t - to l � r.

If we fix A and let the first approximation be xo(t, A) = xo , then the choice of r guarantees that the sequence of successive approximations xj + 1 = .xI;. Xj ' j = 0, 1 , 2, . . . , converges uniformly on I t - to l � r to x = xC!, A), a unique solution of (4) satisfying x(to , A) = Xo .

Furthermore, for s > m we have

O < IX < l ,


for I t - to l s r and I I A - Ao l l s p. This shows that the sequence of successive approximations converges uniformly in A as well, and hence x = x(t, },) is a continuous function of A as wel l .

To conclude this chapter we wish to discuss briefly two results related to the dependence of solutions on parameters or on initial conditions. The method of proof is merely indicated .

First of all suppose we consider the equation (4), where t and A are scalar quantities and x = (XI ' . . . , xn) is a vector variable. Given (to , Xo , ,1.0), suppose that we know the solution X = x(t, ,1.0) satisfying x(to , ,1.0) = xo , and for A near ,1.0 we wish to find an approximation to the solution x = x(t, A) satisfying x(to , A) = Xo .

For 1 ,1. - ,1.0 1 sufficiently small we may use the approximation

x(t, A) = x(t, Ao) + (A - Ao)y(t) ,

where

( ox(t, A) I y t) = --0,1. ), = ).0

(5)

This derivative will exist, for instance, if I i s continuously differentiable with respect to the variable A.. Substitute the expression (5) for x(t, A) in (4) and differentiate with respect to A and set A = Ao . This shows that y(t) satisfies the relation

y = A(t)y + B(t),

where

A(t) = (Of i) OXj ). = ).0

x = x(t .).o)

(Of1 Oln) B(t) = ::l 1 , . . . , -::;-;-UII. 011. ). = ),0

x = x(t ,).o) This i s the variational equation for the differential equation (4), and evidently is a first-order nonhomogeneous system with i nitial conditions y(to) = 0, since x( to , A) = x(to , ,1.0) ' If we can solve or obta in an approximation to y(t) in some neighborhood of to , then the relation (5) will give us an approximation to the solution x = x(t, A) .

Example : Let 11 = 1 and consider the equation .x = AX2 + t. Letting )'0 = to = Xo = 0, we have

x(t, Ao) = x(t, 0) = ( 2/2, x(O, 0) = O.

6 . 3 The Dependence o f Solutions on Parameters 1 39

Furthermore, of/ox = 2XA and of/oA = X2 and the variational equation then becomes

yeO) = 0,

whose solution is y(t) = t 5/20. Then for A near zero we have the approximation

I A x(t, A) = 2" t 2 +

20 t 5

•

Finally, suppose we are given two differential equations with initial conditions

x = f(t, x),

x(to) = Xo ,

y = get, Y),

y(to) = Y I '

where we assume thatf and 9 are defined in some domain B in Rn + I , and

sup I I f( t, x) - get, x) 1 1 < e, ( t, x) i n B,

where e is sufficiently small. Suppose that we know the solution of the second equation and that I l xo - YI I I < b with b sufficiently small. We wish to obtain an estimate of the error obtained by replacing the first equation by the second.

We will assume that f(t, x) is continuous and satisfies the Lipschitz condition

I lf( t, X l ) - f(t, x2) 1 1 s k I l x i - x2 11

for (t, XI ) ' (t, X2) in B. Our assumptions lead to the following estimates :

I l x( t) - y(t) 1 1 s I l xo - Y I I I + l it [f(S, xes»� - g(s, Y(S»] ds l l . t

S b + J I l f(s , yes»� - g(s , y(s» l l ds to

+ { l l f(s , xes»� - f(s , y(s» 1 1 ds to

s b + e I t - to l + k { l l x(s) - y(s) 1 1 ds . to

If we let

M = sup I l x(t) - y(t) l l , I t - to l ::;; r,

with r sufficiently small, we have

I l x(t) - y(t) I I ::;; fJ + e I t - to l + kM I t - to l

for I t - to l ::;; r. Now substitute this expression on the right side of the above

inequality. This gives the relation

I I x(t) - y(t) 1 I k i t - t 1 2 e I t - t 1 2

< fJ + kfJ I t - t I + e I t - t I + e 0 + M

0 - 0 0

2 ! 2 !

Repeating the substitution n times, we have the relation

I I x( t) - y( t) ! 1

n - 1 ki l t _ t li n k i- 1 I t - t l i kn i t - t i n ::;; fJ.L . , 0 + e .

L . , 0 + M , 0 ) = 0 J . ) = 1 J . n .

for I t - to l ::;; r. The last term on the right goes to zero uniformly as n approaches 00, and th is leads to the estimate

e I I x(t) - y(t) 1 I ::;; fJ exp(k I t - to D + k [exp(k I t - to ! ) - 1 ] .

Example : W e consider the pair o f differential equations

x = J(t, x) = 1 + x 2 + (2 ,

y = g( t, y) = 1 + y2 ,

with the same initial conditions x(O) = yeO) = O. Hence fJ = O. If we let B be the domain

B = {( t, x) l l t l < !, I x l < I } , then

I J(t , x) - get, x) 1 = I t2 1 < -(6 = E,

and

I J( t , x) - J( t, Y) ! = I x2 - / 1 < 2 lx - Y I ·

Problems 1 4 1

Hence k = 2 . The solution o f the second equation i s y(t) = tan t, and an estimate of the error in replacing the first equation by the second is given by

I x( t) - tan t l � -iz{e1 /2 - 1J = 0.0203,

valid for I t I � r < t. The estimate above can also be used to measure the change in a solution due to a change in the initial conditions : in this case e = O.

Problems

1. Use the method of successive approximations to find the first three

approximations to solutions of the following equations.

(a) x = 12 - X, x(O) = 1 .

(b) x = t2 - x, x( 1 ) = 2.

(c) x = 1 + 12 + xz, x(O) = O.

2. (a) Using the inequality

00 I Ix - xj l l ::::: 2: I lxS + l - x, l l ,

s = j

(d) x = x2 - I, x( 1 ) = 1 .

(e) x = t + y, x(O) = 2.

y = t - X2, yeO) = 1 .

show that an upper bound for the error in stopping at the jth succes

sive approximation to a solution is given by

m (nkr)J + l I lx - Xj l ie :::::

nk e·kr

(j + I ) ! '

where m , n, k and r are as in the proof of Theorem 1 .3 . 1 .

(b) For I t I < k and Ix i < !, show that the error in stopping at the

third approximation of equation I (c) is less than 1 .08 x 1 0 - 3•

3. Given the scalar equation

x = [(t, x), X(/o) = Xo ,

and a positive integer N, some simple techniques for obtaining in

N steps an approximate value XN of x(to + T), T > 0, are as follows.

(i) Euler-Cauchy method

Xn + 1 = x. + h[(t. , x.), n = 0, 1 , . . . , N - 1 .

(U) Taylor series method of order 2 :

h 2 x. + 1 = X. + hf(t. , x.) + "2 [f,(t. , x.) + fx(t. , x.)f(t. , x.»),

n = 0, 1 , . . . , N - 1 .

(iii) Modified Euler method :

X. + 2 = x. + 2hf(t. + 1 , x. + I ), n = 0, 1 , . . . , N - 2,

where x I is obtained by some other method. The number x. is the

approximate value of x(t.), where 1. + I - I. = h = TIN.

(a) Use the methods (i) and (iii) with X I obtained using (ii) and

N = 1 0 to approximate the value x(1 ) of the solution of x = x, x(O) = 1 . (b) Use the methods (i) and (ii) with N = 1 0 and N = 5 respectively

to approximate the value x(1 ) of the solution of x = t2 + x, x(O) = 1 . (c) Use the methods (i) and (iii) with X I obtained using (ii) and

N = 5 to approximate the value x(1 . 50) of the solution of x = 1 + xl, x( 1 .45) = 8.238 . In all cases compare the results with the actual values of the solution.

4. Let f(t, x) satisfy the hypotheses of Theorem 1 . 3 . 1 in the strip

- 00 < t < 00 , a ::;; x ::;; b, and suppose that f is periodic in t with

period T. If

f(t, a) > 0, f(t, b) < 0, - 00 < t < 00 ,

show that this implies that the equation x = f(/, x) has a periodic

solution of period T. (Hint : First show that any solution x(t) satisfying

x(O) = Xo , a ::;; Xo ::;; b, is defined for 0 ::;; t ::;; T and takes on values

between a and b. Then show there exists such a solution satisfying

x(O) = x(T) and that its periodic continuation is the required solution.)

5. Show that if f(t, x) has continuous partial derivatives up to order m, then any solution of x = f(t, x) has continuous derivatives up to order

m + l .

6. Given the equation

x = x + A(t + X2), I A I < 0. 1 ,

and let the solution x(t, Ao) = x(t, 0) satisfy the initial condition

x(O, 0) = 1 . Find the solution of the variational equation, and use it

to obtain an estimate for 0 ::;; t < t of the difference between the

solutions x(t, 0) and x(t, A), where x(O, A) = 1 .

Problems 143 7. Given the equation

x = t2 + e' sin x, x(O) = 0 = xo ,

estimate the variation of the solution for 0 :::;; t < t if Xo is perturbed

by 0.0 1 .

8 . Find the error i n using the approximate solution

for the equation

.x + tx = 0,

where I t I :::;; !. x(O) = 1 , x(O) = 0,

A P P E N D I X

A Series Solutions of

Second-Order Linear Equations

In general, given the nth order homogenous equation

( 1) there is no method of finding a set of fundamental solutions, and hence a solution, explicitly . The following approach is often useful. We assume that a solution of ( l ) can be expressed in the form

00 yet) = ( t - toY I rxit - to)

j, j = O

rx j constant, (2)

where the power series in the expression is convergent in some neighborhood I t - to l < a of to. In the case r < 0, then, the expression will be valid in the deleted neighborhood 0 < It - to l < a.

Differentiating the expression by terms, and substituting it in ( 1 ) will hopefully give us a solvable recurrence relation for each rxj in terms of its predecessors, and specifically in terms of the first n

rx/s ; hence rxj = cp)rxo , " " rxn - I ), j � n. The expression (2) is then a formal solution, and we must verify the convergence of the series to assure us that it is in fact a solution. The arbitrary constants rxo , . . . , rxn - 1 will be prescribed by given initial values of the solution.

The described method is especially fruitful in the case n = 2-that is, for second-order linear equations-and we proceed to discuss

Appendix A 1 45

it in detai l . For the sake of generality, we will assume that all functions concerned are complex-valued functions of the complex variable z. Our equation then becomes

d2 w dw dz2 + p(z) dz

+ q(z)w = 0,

and we will attempt to find a solution w(z) of the form

co w(z) = (z - zoY L: an(z - zot, n = O

(3)

(4)

where the power series converges in some neighborhood of the point Zo ' We require some preliminary definitions.

D E F I N I T I O N : A function <p(z) i s said to be analytic at z = Zo if it has a Taylor's series expansion <p{z) = L::'= o bn(z - Z ot, which converges in some neighborhood of z = Zo .

D E F I N I T I O N : A function O{z) is said to possess a pole of order k at Z = Zo if O{z) = <p{z)(z - zo) - \ where <p{z) is analytic at z = Zo o If either of the functions p{z), q(z) in (3) is not analytic at z = zo ,

then we say that z = Zo is a singular point of (3) . The class of singular points that we will consider is defined as follows.

D E F I N I T I O N : The singular point z = Zo i s said to be a regular singular point of (3) if p{z) and q(z) possess at most a pole of order 1 and 2 respectively at z = Zo . D E F I N I T I O N : The point z = Cl) is said to be a regular singular point of (3) if, under the transformation z = I /�, the point � = 0 is a regular singular point of the transformed equation.

Remark : We may easily verify that the transformed equation is given by

1 46 Appendix A

From the above definitions it fol lows that if Zo is a regular singular po int of (3), we may write the equation in the form

d2w P(z) dw Q(z) - + -- - + w = O (3 ') dz2 z - Zo dz (z - ZO)2 '

where P(z) and Q(z) are analytic at Z = Zo . Therefore we may write 00 00

P(z) = L Pn(z - zot, Q(z) = L qn(z - zot, n = O n = O

where both series converge i n I z - zo l < a, a > O . To simplify computations we assume that ao = 1 in (4) .

T H E O R E M A. I . If Zo is a regular singular point of (3), then there exists a solution of the form

w(z) = (z - zo)' [ 1 + n�

laiz - zo>"] .

and the above expression is valid in 0 < I z - zo l < a.

Proof Recalling that

we substitute the expression· (5) for w(z) in (3') , which gives us

(5)

(z - zoY{ r2 + r(po - 1) + qo + n�l [aner + n)(r + n - 1) + rPn n - 1 n - 1 ] } + k�O

an -ir + n - k)Pk + qn + k�O

Qn - k qk (z - 20)n = O.

Let F(r + n) = (r + /1)2 + (po - \ )(r + n) + qo , 11 = 0, 1 , 2, . . . , and set k = 0, and the above expression can be written

where IjJ n is a term depending on a I > • • • , an _ l ' Pk ' and qk , k = 1 , • • • , f7 .

Each coefficient must be equal to zero, and hence

F(r) = r2 + (Po - 1 )r + qo = O. (7)

Appendix A 1 47

This is called the indicial equation of (3') at Z = Zo . Let rl and r2 be the two roots of F(r) and let s = rl - r2 be such that Re(s) 2 o. Since F(rl ) = 0 and Po - 1 = - (rl + r2) = - 2rl + s, we have for all 11

F(r ! + n) = F(r l ) + 2nr l + n2 + (Po - 1)n = n2 + n(po - 1 + 21" 1 ) = n(s + n) =1= o.

Therefore we can solve the recurrence relation an F(rl + n) + t/ln = 0 for an i n terms of a I , . . . , an _ 1 , Pk , and qk , k = 1 , . . . , n ; hence (5) with r = rl represents a formal solution of (3'), and hence of (3).

To prove that i t represents an actual solution, we need only show that the series I:= 1 an(z - zot has a positive radius of convergence. Then, since the only singularity of p(z) and q(z) in I z - zo l < a occurs at z = zo , by an analogue of Theorem 2.2. 1 the solution m(z) is defined in 0 I such that

IPn l , I qn l , and I r lPn + qn l are less than Klrxn for all n . Since In + s l 2 11 for any n, we have

I t/l l I lr l P I + q l

l K

l a l l = F(r l + 1 ) = s + l � -; . Suppose that l an l � K nlrxn for /1 = 1 , . . . , m - 1 ; then by computing t/lm we have

l am l = I t/lm I F(r l + 111 ) [I;: II l a m - d I r l Pk + qk l + I r l Pm + qm l

+ I;:l ( m - k) l am - I I I Pk l ] � -----------------------=�--------------�

/11 I s + m I

Hence l all l � K nlrxn for a l l 11, which impl ies that I:= 1 an(z - zot has a rad ius of convergence at least as large as rxlK > O. This proves the theorem.

1 48 Appendix A

We now discuss the possibility of obtaining another solution, corresponding to the root r2 of the indicial equation. In addition, the two solutions constructed will be l inearly independent, and hence all solutions of (3) can be obtained .

C O R 0 L L A R Y. If Zo is a regular singular point of (3), then corresponding to the root r2 of the indicial equation there exists another solution of (3) of the form

w(z) = (z - ZoY2 r 1 + n�lbn(Z - zot] ,

or of the form 00

w(z) = wl (z){3 10g(z - zo) + (z - ZO)'2 L bn(z - zot, n � O where wl (z) is the solution corresponding to the root r l and {3 is a constant. The expressions are valid in 0 < I z - zo l < a.

Proof Two cases occur depending on the value of s = rl - r2 . Case 1 . s "l= 0, positive integer. In this case F (r 2 + n) =

n( - s + n) "1= 0, and we can solve the recurrence relation bn F(r2 + n) + Qn = 0, where Qn corresponds to the previous I/In . If A =

sUPn ", I I I - s/n l - I , it can be shown that for 0 < a < a we have I bn l :::; (MAY/an for all n, with M defined as before, and the result follows.

Case 2. s = 0, or s = positive integer. If s = 0, then rl = r2 , so noth ing new is gained ; if s = m , then F(r2 + m) = 0 and the recurrence relation cannot be solved . We proceed by the method of reduction of order and assume there exists a solution of the form

w2(z) = w l (z) j"<P(u) du, where w j (z) is the solution corresponding

to r = r l '

Substitution in ( 3 ') leads to the equation ( dW I ) d<p P(z) 2 dz - + -- + -- <p = O. dz z - Zo W I

Appendix A

The term in parentheses can be expressed in the form

- (Po + 2r, ) ----'''--'--------'''- + <P2(Z) , z - Zo

where <P2(Z) is analytic at Zo , and hence

<p(z) = exp{f [ - (�o_

+z:r , )

+ <Pz(U)] dU } = (z - zo) - po - 2r ' <P3(Z),

where <P3(Z) is analytic at Zo . Since -Po - 2r, = - s - 1 ,

00 <p(z) = (z - zo) - . - 1 I f3n(Z - Zo)",

n = O

and now integrating <p(z) we have (if f3 = f3. i= 0)

where <P4(Z) is analytic at z = Zo . We write

n�o biz - zo)" = <Piz) { 1 + n�) aiz - zo)"} ,

and since - s + r) = r2 ' we fina lly have

00 w2 (z) = w ) (z)f3 log(z - zo) + (z - ZO)'2 I bn(z - zo)n,

n = O

1 49

which is the required result . Tf fJ = fJs = 0, then wz{z) is of the form given i n Case I . Note that if s = 0, the logarithmic form always appears .

Tn summary, to find a solution of (3) that is valid near a regular point Zo , we solve the i ndicial equation to find r) , then substitute in (3) a series

(z - zoY ' { 1 + J,an(z - zo)n}

1 50 Appendix A

The recurrence relation is solvable and we can find the an explicitly. To find a second l inearly independent solution, we must consider the value of s. The examples below illustrate several cases.

We now briefly discuss the case in which the coefficients p(z) and q(z) in the equation (3) are analytic at z = Zo .

D E F I N I T I O N : The point z = Zo is said to be an ordinary point of (3) if p(z) and q(z) are analytic at z = Zo .

It follows then that p(z) and q(z) are expressible in power series

00 00 p(z) = L bn(z - zo)n, q(Z) = I Cn(Z - zot,

n = O n = O

which converge i n some neighborhood o f z = Zo , say for I z - zo l < a. If we substitute the series expression for p(z) and q(z) in (3) and assume a solution of the form

00 w(z) = L an(z - zot,

n = O

we are led to the recurrence relation n

(n + 2)(n + l )an + 2 = - L [(k + l )bn - k ak + l + Cn - k akJ · k = O

Letting ao and a 1 be arbitrary, w e can solve for an , n � 2 , i n terms of them.

It can be shown that the resulting power series for w(z) converges for Iz - zo l < a, and therefore represents a solution . In most cases, since ao and al are arbitrary, we can obtain two linearly independent solutions. In any case, since ao = w(zo) and a1 = w'(zo), we can determine the solution satisfying any given initial conditions.

Examples (a) The equation

d2w dw 2z - + - - w = 0

dz2 dz

Appendix A

can be written in the form

d2w 1 /2 dw - z/2 - + - - + -- w = O. dz2 z dz Z2

1 5 1

Therefore z = 0 i s a regular singular point, Po = t, qo = 0, and the indicial equation is , 2 - !, = o. Its roots are 'I = ! '2 = 0 ; hence s = ! and s i= 0, positive integer.

Substituting the series

w W I (z) = I dn zn + ( 1 /2 ) o

in the original equation gives

ro ro

I 2(n + !)(n - !)an zn - ( 1 /2 ) + L (n + !)an zn - ( 1 /2 ) o 0 w

- " a zn+ ( 1 /2 ) = 0 L. n , o

and combining the first two series and shifting indices in the last gives

ro I [11(2n + l )an - an _ l ]zn - ( 1 / 2 ) = o.

1

Therefore

2an - 1 a = --"'----=---n (211 + 1 )(211 ) or

2nao a = ---=--n (211 + 1) ! '

and therefore

ro 2n w (z) = a 2 1 / 2 " Zn. 1 0 i' (2n + I ) !

11 � 1,

Proceeding in a s imilar manner for '2 = 0 gives a second l inearly independent solution,

w 2n W2(Z) = bo � (211) ! zn ,

where ao and bo are arbitrary constants .

1 52 Appendix A

(b) The equation

d2w dw Z -

d 2 + (3 + Z3) - + 3z2w = 0

z dz


d2w 3 + Z 3 dw 3z3 - + -- - + - w = O. dz2 Z dz Z 2

Therefore z = 0 i s a regular singular point, Po = 3 , qo = 0 , and the indicial equation is r2 + 2r = O. Its roots are r1 = 0, r

2 =

- 2 ; hence s = 2, a positive integer. In this case i t i s advantageous to use r = r

2 , so let

00 w(z) = L an z

n - 2 . o

Substituting in the original equation and shifting indices gives

00 - a 1 z + L [en - 2)nan + (n - 2)an _

3]zn - 3 = O.

3 Therefore a 1 = 0 and an = ( - an - 3

)!n, which implies that

a 1 = a4 = . . . = a3n + 1 = . . . = 0,

and

( - 1)nao ( - 1)nao a

3n =

(3n)(3n - 3) · · · (3) =

3nn !

( - 1 )"a2 a

3n + 2 =

(3n + 2) . . . 8 . 5 '

where Go and ao are arbi trary . Then we have [ 00 ( - 1 )" J [ 00 ( - 1)" J w(z) = Go Z - 2 + L -n- z3n - 2 + G2 1 + L Z3n

1 3 n ! 1 (3n + 2) · . . 8 ·5

and the bracketed series represent a pair of l inearly independent solutions corresponding to the roots of the i ndicial equation. Evidently this is the nonlogarithmic case .

Appendix A 1 53

(c) The equation

d2w dw (z - 1)

dz2 + z

dz + w = 0


d2w 1 + (z - 1 ) dw z - 1 - + - + w = o. dz2 z - 1 dz (z - 1)2

Therefore z = 1 is a regulaT singular point, Po = I , qo = 0, and the indicial equation is r2 = 0 ; hence s = 0, a logarithmic case.

F or this case it i s convenient to assume a solution of the form

00 w(z) = (z - 1 ), + L an(z - It + r, 1

where r is indeterminate. Writing the equation in the form

d2w dw (z - 1 ) -2 + [ 1 + (z - 1 )] - + w = 0,

dz dz

and substituting w(z) as given above leads to the relation

00 + L [en + r)2an + (n + r)an _ I](z - 1 )n + r- 1 = 0.

Therefore

and

- 1 a l = -- , r + 1

oan = ( _ l)n + l or

2

- an - 1 ( - It a = -- = -----'---'------n n + r (n + r)(n - l + r) " ' ( l + r) '

l 1 1 1 ] -- + + . . . + --n + r n - 1 + r l + r

(n + r)(n - 1 + r) " . ( 1 + r)

1 54

It follows that

I ( _ I)n an = --, - , r = O n .

oan l = ( - l )n + 1Hn

or r = O II ! '

where Hn = Ii 1 1k . The two l inearly independent solutions of the equations are then given by

00 ( _ 1 )n w 1 (z) = 1 + I -- (z - 1)",

1 n !

00 ( - l )" + l H W2 = w 1 (z)log(z - 1) + I n (z - 1 )n.

1 n !

For the case s = positive integer, logarithmic case, we should assume a solution of the form

00 w(z, r) = (z - zo)'(r - 1'2) + L anCz - zo)" + r,

1

then substitute and evaluate an = an(r) . The two l inearly independent solutions are given by

Finally, it should be noted that solutions need not have s ingularities at a regu lar si ngular point. For instance, the point z = Zo is a regular singular point of the equation

d2w - 2 dw 2 - + -- - + w = o dz2 z - Zo dz (z - ZO)

2 '

but a fundamental pair of solutions is given by

which are analytic at z = Zo .

Linear Systems with

Periodic Coefficients

In this appendix we will discuss the l inear system

x = A(t)x,

A P P E N D I X

B

(1) where x = (X l ' . . . , xn) and A(t) = (aij(t)) is a continuous periodic matrix defined on - 00 < t < 00 . Therefore there exists a nonzero number T such that

A ( t + T) = A(t), - oo < t < oo .

It does not necessari ly fol low that there exists a nontrivial solution x(t ) of ( I ) having period T.

Example : For the equation

x = (cos2 t)x, X = xU) a scalar function , we have

a( t + 2n) = aCt) . Any nontri vial solution is of the form r t s in 2t] x(t) = Xo exp :2 + -4 -

and is not periodic.

1 56 Appendix B

A basic result (Floquet's theorem) describing the properties of solutions of ( 1 ) i s the fol lowing.

T H E O R E M B . l . Given the system ( I ) with A(t) a continuous periodic matrix with period T, then there exists a nonzero constant A (real or complex) and at least one nontrivial solution x( t ) of ( 1 ) having the property that

x(t + T) = h(t),

Proof Let

- oo < t < oo .

(t) = ((0) =I- 0, and

det <1>(t) = det <1>(0) exp [( tr A(s) ds l (2)

by the results of Chapter 2. It follows that det <1>(t + T) =I- 0 and hence <1>( t + T) i s also a fundamental matrix, and its columns therefore form a fundamental system of solutions.

Hence there exist constants c jk , j, k = 1 , . . . , n such that n

<Pik(t + T) = L Cjk <Pij( t) , j = 1

i, k = 1 , . . . , n ,

or equivalently there exists an n x n constant matrix C such that <1>(t + T) = <1>(t)C, - oo < t < oo .

If we let t = 0, then (2) impl ies that

det C = exp [ ( tr A(s) dS] =I- o.

Let A be a characteristic root ( real or complex) of the matrix C-that is, a root of the characteristic polynomial det(C - AI) . Then det C =I- 0 implies A =I- 0 and furthermore there exists a nonzero vector rx such that Crx = ).rx .

Appendix B 1 57

Consider the solution x(t) = <D(t)a of ( 1 ) . Then for all t we have x(t + T) = <D(t + T)rx = <D(t)Crx

= <D(t)Aa = A<D(t)a = Ax(t), and conversely for any solution having this property, A must be a characteristic root of C. If )' 1 ' . . . , Am ' 1 S m S n, are the distinct characteristic roots of C, then there are at least m solutions of ( 1 ) satisfying the conclusion o f the theorem . This completes the proof.

It should be noted that if net) i s any other fundamental matrix, then there exists a constant matrix A , with det A ::j:. 0, such that net) = <D(t)A . Since <D(t + T) = <D(t)C, we have

n(t + T) = <D(l + T)A = <D(t)CA = n(t)A - 1 CA .

From l inear algebra, w e know that the characteristic roots o f C and A - I CA are the same, and therefore the numbers A I , . . . , Am ' 1 s m s n, are independent of the choice of a fundamental matrix .

D E F I N I T I O N : The d istinct characteristic roots A I ' . . . , Am of the matrix C are called the characteristic factors or multipliers of the system ( I ) . The numbers rl , • . • , r m defined by the relations A i = eriT, i = 1 , . . . , m , are called the characteristic exponents of the system. We then have the following corollary to the previous theorem. C O R O L L A R Y. The system ( 1 ) has a periodic solution of period T if and only if there is at least one characteristic factor equal to unity.

From the definition it follows that the characteristic exponents are determined up to mu l tiples of 2rr.ijT, where i = J - J . If A; is a

characteristic factor of ( J ) and x(t ) is a corresponding solution satisfying x( t + T) = A ; x( t) , then write x(t ) in the form

x(t) = p( t)er " . Then we have

pet + T)er iU + TJ = Ai P( t)er" , and by the definit ion of r i , it fol lows that p ( t + T) = p( t ) . Thus we have the following resu l t .

1 58 Appendix B

T H E O R E M B.2. If r l , • . . , r m are the characteristic exponents of the system ( l ) , then there are at least m solutions of the form

j = 1 , . . . , m,

where the functions p ;(t ) are periodic with period T.

Finally, suppose we take multipl icities into account and denote by A I , . . . , All the characteristic roots of the matrix C, and by rl , • • • , rn the corresponding exponents . If we choose the fundamental matrix (t) so that <1>(0) = I, then C = (T). From l inear algebra and the previous discussion follows the relation

exp(r l + . . . + rn)T = Al . . . An = det (T)

= det C = expU: tr A(s) dSj , and therefore

T

' 1 + . . . + I"n == T - 1 f tr A(s) ds o

(mod 2IT i /T) .

Example : Consider the second-order l inear equation ( H il l ' s equation)

x + p(t)x = 0,

where p(t) i s rea l-valued , continuous, and periodic with per iod T, say T = IT . We choose a fundamental system of solutions

x ; ( t) , x2( t) satisfying

X z (O) = 0,

and hence the i r Wronskian equals I . The preced ing resu lts imply that

si nce (' 0 01 )

A(t) = ( ' , - P / ) and

Appendix B 1 59

The characteristic roots A I ' )'2 are therefore roots of the equation

which impl ies that }' I Az = I . The characteri stic exponents r I ' r 2 satisfy the relation

(mod 2i), so we may define (up to an in teger multiple of 2) the n umber r

sat isfying

From Floquet's theorem and i ts corollaries it fol lows that

(i) i f Al :f= Az , then H i l l ' s equation has two l i nearly independent solut ions ,

where P I ( t ) and pz ( t ) are periodic with period 11: ;

( ii) i f 1. 1 = )'z , then H i l l ' s equat ion has a nontriv ia l solut ion,

which i s per iodic with period 11: (when 1.1 = Az = I ) or period 211: ( when ). 1 = I.z =-= - J ) . Tn the last case the solut ion i s of the form e i Z tp( t ) or ei tp( t ) , where p( t ) i s period ic with period 11: .

T h e difficu lty i n apply i ng t h e above resu l ts i s that w e m ust have

information about the matr ix C, or eq u ivalently, a bout a funda

mental system of solut ions of ( I ) , to be able to determ ine the char

acter ist ic exponents . But for certa in systems and i n part icu lar for H i 1 l ' s eq uat ion , cond it ions on A ( t ) have been given that guarantee that there exist stable, unstable, or osci l latory so lut ions .

We conclude by giv ing a resu l t for the nonhomogeneous system

x = A(t)x + B( t) , (3)

where A ( t ) and B( t ) are conti nuous and period ic with period T.

1 60 Appendix B

T H E O R E M B .3 : The system (3) has a periodic solution of period T for every B(t) if and only if the corresponding homogeneous system has no nontrivial solution of period T.

Proof A periodic solution x( t ) with period T satisfies the relation x(O) = x(T) ; conversely, if there exists a solution satisfying this relation, then i t must be periodic. For if x(t ) is a solution , then yet ) = x(t + T) is also a solution since A(t) and B(t) are periodic. The relation x(O) = x(T) implies x(O) = yeO), and by uniqueness this implies x(t) = y(t) for all t; hence x(t) i s periodic.

By the results of Chapter 2, the solution x(t ) of (3) satisfying x(O) = Xo i s given by

. 1 x( t) = (t)xo + j ( t)- l (s)B(s) ds, o

- 00 < t < 00 ,

where (t) i s a fundamental matrix o f the corresponding homogeneous system and <1>(0) = I. From the previous remarks, the existence of a periodic solution is therefore equivalent to the relation

T x(O) = Xo = x(T) = (T)xo + f (T)- I(S)B(s) ds , o

or equivalently, for every B(t) we can solve T

[J - (T)Jxo = f (T) - I (S)B(s) ds. o

But the last equation will have a solution Xo if and only if det [1 - (T)] "# 0, which i s equivalent to the assertion that the equation (T)xo = Xo has only the trivial solution Xo = O .

However, the solutions of the corresponding homogeneous system x = A(t)x are given by <p(t) = (t)xo , and therefore the relation

<peT) = ( T)xo = <p(0) = Xo

can only be satisfied by the trivial solution <pet) == O. Again by the remarks above, this is equivalent to the assertion that the homogeneous system has only the trivial solution as a periodic solution of period T, and this proves the theorem.

References

General References and Introductory Works

1 . G. Birkhoff and G. Rota, Ordinary Differential Equations, Ginn,

Boston, 1 962.

2 . F. Brauer and J . A . Nohel , Ordinary Differential Equations, Benjamin,

New York , 1 967. 3 . J . L. Brenner, Problems in Differential Equations, 2nd ed. , W. H.

Freeman and Company, San Francisco and London, 1 966.

4. E. Coddington, An Introduction to Ordinary Differential Equations, Prentice-Hall , Englewood Cl iffs, N.J . , 1 96 1 .

5. E. Coddington and N . Levinson, Theory of Ordinary Differential Equations, McGraw-H ili , New York, 1 95 5 .

6. W. Kaplan, Ordinary Differential Equations, Addison-Wesley,

Reading, M ass. , 1 958 . 7. W. Leighton, Ordinary Dif/erefltial Equations, 2nd ed. , Wadsworth,

Belmont, Cal if. , 1 966.

8. K . Yosida, Lectures on Differential and Integral Equations, Inter

science, N ew York , 1 960.

References Emphasizing Stability Theory

1 . R. Bellman, Stahility Theory of Ordinary Differential Equations, McGraw- H i l i , N ew York, 1 95 3 .

1 62 References

2. L. Cesari , Asymptotic Behavior and Stability Problems in Ordinary Differential Equations, 2nd ed. , Academic Press, Ne w York, 1 963.

3 . W. Coppel, Stability and Asymptotic Behavior 0/ Differential Equations, Heath, Boston, 1 965.

4. W. H ahn, Theory and Application 0/ Liapunov's Direct Method (translated by H . Hosenthien and S. Lehnigh) , Prentice-Hall , Engle

wood Cliffs, N.J . , 1 96 3 .

5 . A . Halanay, Differential Equations, Academic Press, New ,\ork, 1 966.

6 . W. Hurewicz, Lectures 011 Ordinary Differential Equations, M . LT.

Press, Cambridge, M ass. , 1 95 8 .

7. J . LaSal le a n d S. Lefschetz , Stability by Liapullov's Direct Method with Applications, Academic Press, New York, 1 96 1 .

8 . S . Lefschetz, Differential Equations: Geometric Theory, 2nd ed. ,

I nterscience, New York , 1 96 3 .

9 . L. Pontryagin , Ordinary Differential Equations (translated b y L .

Kacinskas a n d W. Counts) , Addison-Wesley, Reading, Mass. , 1 962.

Index

A Annihilator method, 5 3 Approximate solutions, 1 3 8-1 4 1 Asymptotic stability

of an isolated critical point, 7 1 for l inear systems, 8 1 , 96--97 for nonlinear systems, 86, 99- 1 04 of a solution, 92

A utonomous system, 68-72 critical point of, 70 phase space of, 7 1 , 8 1 -83 trajectory of, 69

B Bessel 's equation , 39 (Problem 9)

c Characteristic exponent , 1 57 Characterist ic factor (mult iplier), 1 57 Characteristic polynomial

of a matrix, 59 of an n th-order l inear equat ion , 44 stability of, 57

Continuation of solutions, 1 30- 1 34 Contract ion mapping, 1 26, 1 30

fixed point of, 1 24, 1 30 Critical point

center, 77, 86 improper node, 80 node, 7 5 , 86 proper node, 79, 86

saddle point, 76, 86 simple, 84 spiral point (focus), 78, 8 5

Cycle, 82

o Dependence of solutions on parameters,

1 36- 1 39 Differential equation, 4-8

normal form, 5 order of, 4 reduction to first-order system, 6 solution of, 5

D irection field, 7

E Euler equation , 66 (Problem 5) Euler-Cauchy method, 1 4 1 (Problem 3) Existence of solutions, 1 2 1 - 1 30

F First-order l inear systems, 1 5-28

constant coefficients, 58-65 characteristic polynomial, 59

nonhomogeneous case, 24-28 periodic coefficients, 1 55-1 60

Floquet's t heorem, 1 56 Fundamental matrix, 2 1 Fundamental system o f solutions

for first-order l inear systems, 1 9 constant coefficients, 62

1 64

for nth-order linear equations, 30 constant coefficients, 48, 50

G Gronwall 's i nequality, 95

H Hill's equation, 40 (Problems 1 2, 1 3),

1 58- 1 59

I Indicial equation, 1 47 Initial values, initial condition, 8 Integral curve, 7

L Legendre's equation, 39 (Problem 1 0) Liapunov function , 1 08 Liapunov's direct (second) method,

1 05-1 I I Limit cycle, 82 Linear asymptotic equilibrium, 1 1 9

(Problem 6) Liouville's formula, 22 Lipschitz condition, 1 29

M Maximum i nterval of existence, 1 0- 1 3 ,

1 3 3- 1 36 for first-order l inear system, 1 7 , 1 3 5-

1 36 Method of successive approximations,

1 2 1 Method o f undetermined coefficients,

53 Modified Euler method, 1 42 (Problem

3)

N nth-order l inear equation, 28-37

constant coefficient�, 4 1 -58 behavior of solutions, 55-58 characteristic polynomial, 44

nonhomogeneous case, 3 3-37 constant coefficients, 5 1 -55

o Oscillatory solution, 1 1 4

Index

p Periodic solutions

existence of, for l inear systems, 1 55-1 60

Phase space, 7 1

R Rectil inear trajectory, 75 Regular singular point, 1 45 Resonance, 65 (Problem 4) Riccati equation, 1 4 (Problem 8), 37

(Problem 2) Routh-H urwitz criteria, 57

s Second-order linear equation

behavior of solutions, 1 1 I - 1 1 7 ordinary point of, 1 50 series solutions of

near an ordinary point, 1 50 near a regular singular point, 1 44--

1 50 singular point of, 1 45

Stability of an isolated critical point, 7 1 for l inear systems, 93-99 for nonl inear systems, 99- 1 04 of a solution, 92 restrictive, 1 1 8 (Problem 4) uniform, 1 02

Sturm comparison theorem, 1 1 4 Sturm separation theorem, 1 20 (Prob

lem 1 0)

T Taylor series method, 1 42 (Problem 3)

u Uniqueness of solutions, 8, 1 0, 1 28

v Van der Pol's equat ion, 1 20 (Problem 9) Variational equation, 1 38

w Wronskian

for first-order l inear system, 2 1 -25 for nth-order l inear equation , 3 1 -3 3

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