Introduction to Ordinary Differential Equations. Academic Press International Edition

434 Pages·1966·18.491 MB·English

by  Albert L. Rabenstein (Auth.)

ACADEMIC PRESS INTERNATIONAL EDITION
Introduction to Ordinary
Differential Equations
ALBERT L. RABENSTEIN
MACALESTER COLLEGE
ST. PAUL, MINNESOTA
(yTTJACADEMIC PRESS New York and London
ACADEMIC PRESS INTERNATIONAL EDITION
This edition not for sale in the United States of America
and Canada.
COPYRIGHT © 1966, BY ACADEMIC PRESS INC.
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PRINTED IN THE UNITED STATES OF AMERICA
PREFACE
This book is intended primarily for undergraduate students of engineering
and the sciences who are interested in applications of differential equations.
It contains a fairly conventional, but careful, description of the more useful
elementary methods of finding solutions. It also contains a number of
topics that are of particular interest in applications. These include Laplace
transforms, eigenvalue problems, special functions, Fourier series, and
boundary-value problems of mathematical physics. The emphasis is on the
mathematical techniques, although a number of applications from ele­
mentary mechanics and electric circuit theory are presented for purposes
of motivation. Finally, some topics that are not directly concerned with
finding solutions, and that should be of interest to the mathematics major,
are considered. Theorems about the existence and uniqueness of solutions
are carefully stated. The final chapter includes a discussion of the stability of
critical points of plane autonomous systems (the approach is via Liapunov's
direct method), and results about the existence of periodic solutions of
nonlinear equations.
The level is such that the material is accessible to the student whose
background includes elementary but not advanced calculus. Because of the
minimum prerequisites, a number of basic theorems have been stated but
not proved. One example of this is the basic existence and uniqueness
theorem for initial value problems. However, the method of successive
approximations, which can be used to prove this theorem and which is
Preface
important in itself, is presented and illustrated in the examples and exercises.
Elementary properties of determinants and theorems about the consistency
of systems of linear algebraic equations are used fairly often. The notion
of a matrix is used on two occasions. The needed results from linear algebra
are presented in a brief appendix, which contains its own set of exercises.
There is sufficient material and flexibility in the book that it can be used
either for an introductory course or for a second course in differential
equations. In a second course, some of the material on elementary methods
of solution might be omitted, or else used for review purposes. Sample
course outlines are given below. There is just about the right amount of
material in the entire book for two semesters' work.
A brief description of the various chapters and some of their special
features is as follows. Much of the material in Chapters 1, 9, and 12 is
fundamental. These chapters deal with the basic theory of single linear
equations, systems of linear equations, and nonlinear equations, respec­
tively. Chapter 2 concerns itself with topics in linear equations which,
although important, are not of such immediate use in applications as those
of Chapter 1. Chapter 3 serves primarily to review the subject of power
series, but from the standpoint of complex variables.
Chapters 5, 6, 7, 10, and 12 are independent of one another, so a fair
amount of flexibility is available in choosing topics. Chapter 11 depends
on Chapter 8, which in turn depends on both Chapters 6 and 7. Chapter 2
can be omitted entirely with little loss of continuity. With students well
versed in the subject of real power series, Chapter 3 can also be omitted.
Only Section 9 of Chapter 4 requires a knowledge of series with complex
terms. Possible outlines for a one-semester introductory course (Course I)
and for a second course (Course II) are given below.
Course I Course II
Ch. 1 Ch. 1, 1.1-1.10
Ch. 2, 2.1 Ch. 4, 4.1-4.8, 4.10
Ch. 3 Ch. 5
Ch. 4, 4.1-4.6 Ch. 6, 6.1-6.8, 6.10
Ch. 5, 5.1-5.2 Ch. 7
Ch. 9 Ch. 8
Ch. 10
Ch. 12, 12.1-12.5
Preface vii
An effort has been made to provide exercises of varying levels of diffi­
culty. Some of the more challenging ones extend the theory presented in
the text, and can be used as bases for classroom presentations if desired
by the instructor. Answers to about half the exercises have been placed
at the end of the book.
The author wishes to express here his appreciation to Professor George
Sell of the University of Minnesota, who reviewed the manuscript and
made many helpful suggestions for its improvement.
St. Paul, Minnesota A.L.R.
1
CHAPTER
LINEAR DIFFERENTIAL EQUATIONS
LI Introduction
An ordinary differential equation is simply an equation that involves a single
unknown function, of a single variable, and some finite number of its deriva­
tives. Examples of differential equations for an unknown function y(x) are
dy
(a) -f. + xn y2=2x
ax 9
d2L *d-l.
(b) 1Z2 ++ e*xÌ7.-y = °·
dx2 dx
The order of a differential equation is the order of the highest order derivative
of the unknown function that appears in the equation. The orders of the
equations in the above examples are one and two, respectively.
The adjective " ordinary " is used to distinguish a differential equation from
one that involves an unknown function of several variables, along with the
partial derivatives of the function. Equations of this latter type are called
partial differential equations. An example of a partial differential equation
for a function u(x,t) of two variables is
d2u d2u du
BF=Ô? + 2Ô- + U-
X
Except for Chapter 11, this book concerns itself mainly with ordinary differen­
tial equations.
A linear ordinary differential equation is an equation of the special form
dny dn~ly dy
floW j^n + *ι(χ) -J^ï + ··' +a -(x)— + a (x)y =/(*), (1.1)
n l n
3
4 I Linear Differential Equations
where the functions a^x) and f(x) are given functions. The functions a^x)
are called the coefficients of the equation. When f(x) = 0, the equation is
said to be homogeneous; otherwise it is said to be nonhomogeneous. It is with
equations of the form (1.1) that we shall be mainly concerned in this chapter.
It will be convenient for us to introduce the operator L by means of the
definition
dn dn~{ d
L = a°(x) ώΐι + °lM ώ^~ι + '*' + a"-l(x) ~dx + a"M' (1,2)
If u{x) is any function that possesses n derivatives, the result of operating on
u(x) with the operator L is the function Lu(x), where
dnu(x) dn~lu(x)
Lu(x) = a0(x) --^d-^j- ++ aa^xM) ~dJ ?n _[ + ··· -f an(x)u(x).
The differential equation (1.1) can now be written more briefly as
Ly=f.
If u{x) and u (x) are any two functions that are n times differentiable,
x 2
and if C and C are any two constants, then
l 2
dm d^.ix) ^ dmu (x)
2
ArCClWl(x) + C2l/2(x)] = Cl "^ + Cl-ώΓ' x*m*n-
As a consequence, the operator L has the property that
L[Cu(x) + C u (x)] = CLu(x) + C Lu {x). (1.3)
{ x 2 2 x x 2 2
This property is described by saying that L is a linear operator. If Wj(x),
u (x), ..., u (x) are functions that possess n derivatives, and if C C , ..., C
2 m l9 2 m
are constants, it can be shown by mathematical induction that
L(Cu + C u + ··. + C u ) = C.Lu, + C Lu + ··· + C Lu . (1.4)
i l 2 2 m m 2 2 m m
By an interval I is meant a set of real numbers of one of the following
types :
a < x <b, a < x < b, a < x < b,
a < x < b, a<x<+ao, a<x<+co,
— oo<x<b, —co<x<b, —oo<x<+oo,
where a and b are constants, with a < b. We shall also use the following
corresponding notations for the nine types of intervals:
(a,b) [a,b) (a,b)
[a,b] [a, +co) (a, +oo)
(-00,/)] (-00,6) (—00, +oo).
I.I Introduction 5
A real solution of a differential equation is a function that, on some interval,
possesses the requisite number of derivatives and satisfies the equation.
Thus a function u(x) is a solution of the linear equation (1.1) on an interval
/ if it is n times differentiable on / and is such that
Lu(x) = f(x)
on /. For example, the function x2 is a solution of the equation
Ly = y" + 3xy' — y = 2 + 5x2
on the interval (— oo, 4- oo) because
L(x2) = (x2)" + 3x(x2Y - x2 = 2 4- 6x2 - x2 = 2 + 5x2
for all x.
By a complex function of the real variable x, we mean an expression of the
form u(x) 4- iv(x), where u(x) and v(x) are real functions and / is the imaginary
unit. Arithmetic laws for complex functions are defined in accordance with
the usual laws for complex numbers. The derivative of a complex function is
defined as
à du(x) . dv(x)
— r[u(x) 4- iv(x)] = —— 4- ι —— .
ax ax ax
Thus the derivative of a complex function is also a complex function.
From now on it will be assumed that the coefficients a^x) in the operator
L are real functions. Then the result of operating on a complex function
u 4- iv with L is
L(u 4- iv) = Lu 4- iLv,
which is also a complex function. If w(x) = u(x) + iv(x) and w (x) =
{ x x 2
u (x) + iv (x) are complex functions, and if C = a + ib and C = a + ib
2 2 i { l 2 2 2
are complex constants, it is easily verified that
L(Cw + C w ) = CLw\ + C Lw . (1.5)
l l 2 2 x 2 2
In fact, for a set of m complex functions vv vt', ..., w , and a set of complex
1? 2 m
constants C C , ..., C , we have
l5 2 w
L(Cw + Cvv 4- ··· 4- C w ) = CLw + C Lw 4- —l· C^Lw^. (1.6)
l l 2 2 m m x x 2 2
A complex function u(x) + iv(x) is a (complex) solution of the differential
equation (1.1) on an interval / if
L[u(x) + iv(x)] =f(x)
on /. Evidently a complex function w = u 4- iv is a solution of the homo­
geneous equation Ly = 0 if, and only if, its real and imaginary parts are real
solutions—that is, if, and only if, Lu = 0 and Lv = 0. If each of the functions
Wj, w , ..., H is a solution, real or complex, of the homogeneous equation
2 w
Ly = 0 on an interval /, and if Q, C , ..., C are any constants, real or
2 m
6 I Linear Differential Equations
complex, then the function
Qw, +C> + ··· + C w
2 m m
is also a solution of the equation on the interval /. This result follows from the
property (1.6) of the linear operator L. It is known as the superposition
principle for real linear homogeneous differential equations.
One particular complex function is of special importance in the study of
certain classes of linear differential equations. This is the complex exponential
function, which we shall define presently. First, however, we define the
complex number ep+iq, where p and q are any real numbers, as
eP+ iq = eP cos q + ieP sin q (1.7)
The number e here is the base of natural logarithms. It should be noted that
when q = 0, the number (1.7) is simply the real number ep. As other special
cases, we have
eiq = cosq + isinq, e~iq = cosq — /sing. (1.8)
Consequently, upon solving for cos q and sin q, we have
eiq + e~iq eiq - e~iq
cos q = 2 , sin q 2/ (1.9)
From the relations (1.7) and (1.8) it follows that
e
The general laws of exponents,
ez>+z> =Zi · ez\ — =eZl"Z2, (1.10)
e
eZl
where z and z are any two complex numbers, follow from the definition
x 2
(1.7) and well-known trigonometric identities. Their verification is left as an
exercise.
Let c = a + ib be an arbitrary complex constant. A complex function of the
form
ecx = eax cos bx + ie ax u n bx (U j)
is called a complex exponential function. A little calculation shows that the
derivative of such a function is given by the familiar formula
d
dx
If a is a positive real number and c is any complex number, we define
(1.12)