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The sole aim of science is the honor of the human mind,
and from this point of view
a question about numbers
is as important
as a question about the system of the world.
-C. G. J . Jacobi

DIFFERENTIAL
EQUATIONS
WITH APPLICATIONS AND
HISTORICAL NOTES
Second Edition

George F. Simmons
Professor of Mathematics
Colorado College

with a new chapter on numerical methods by
JohnS. Robertson
Department of Mathematical Sciences
United States Military Academy

McGraw-Hill, Inc.
New York St. Louis San Francisco Auckland Bogota Caracas

Hamburg Lisbon London Madrid Mexico Milan Montreal
New Delhi Paris San Juan Sao Paulo Singapore Sydney
Tokyo Toronto

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This book was set in Times Roman .
The editors were Richard Wallis and John M. Morriss ;
the production supervisor was Louise Karam.
The cover was designed by Carla B auer.
Project supervision was done by The Universities Press .
R. R. Donnelley & Sons Company was printer and binder.

DIFFERENTIAL EQUATIONS WITH APPLICATIONS
AND HISTORICAL NOTES

Copyright© 1991 , 1 972 by McGraw-Hill , Inc. All rights reserved . Printed in the United
States of America . Except as permitted under the United States Copyright Act of 1976,
no part of this publication may be reproduced or distributed in any form or by any
means , or stored in a data base or retrieval system , w:thout the prior written permission
of the publisher.
2 3 4 5 6 7 8 9 0 DOC DOC 9 5 4 3 2 1

ISBN 0-07-057540-1
Library of Congress Cataloging-in-Publication Data

Simmons, George Finlay, (date).

Differential equations with applications and historical notes I
George F. Simmons.-2nd ed.
em.
p.
ISBN 0-07-057540- 1
I. Title.
1. Differential equations.
QA372.S49
199 1
5 15 '.3�c20

90-33686

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ABOUT THE AUTHOR

George Simmons has academic degrees from the California Institute of
Technology, the University of Chicago , and Yale University . He taught
at several colleges and universities before joining the faculty of Colorado
College in 1962, where he is Professor of Mathematics . He is also the
author of Introduction to Topology and Modern Analysis (McGraw-Hill ,
1963), Precalculus Mathematics in a Nutshell (Janson Publications , 198 1 ) ,
and Calculus with Analytic Geometry (McGraw-Hill , 1985 ).
When not working or talking or eating or drinking or cooking,
Professor Simmons is likely to be traveling (Western and Southern
Europe, Turkey , Israel , Egypt , Russia, China, Southeast Asia) , trout

fishing (Rocky Mountain states) , playing pocket billiards , or reading
(literature , history , biography and autobiography , science , and enough
thrillers to achieve enjoyment without guilt).

vii

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/>FOR HOPE AND NANCY

my wife and daughter
who still make it all worthwhile

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CONTENTS

Preface to the Second Edition
Preface to the First Edition
Suggestions for the Instructor
1

The Nature of Differential Equations. Separable
Equations

1.
2.
3.
4.
5.
6.

Introduction
General Remarks on Solutions
Families of Curves. Orthogonal Trajectories
Growth, Decay, Chemical Reactions, and Mixing
Falling Bodies and Other Motion Problems
The Brachistochrone. Fermat and the Bernoullis

2

First Order Equations

3


Second Order Linear Equations

7.
8.
9.
10.
11.
12.
13.

14.
15.
16.
17.
18.

Homogeneous Equations
Exact Equations
Integrating Factors
Linear Equations
Reduction o f Order
The Hanging Chain. Pursuit Curves
Simple Electric Circuits

Introduction
The General Solution of the Homogeneous Equation
The Use of a Known Solution to Find Another
The Homogeneous Equation with Constant Coefficients
The Method of Undetermined Coefficients


XV

xvii
xxi

1
1
4
10
17
29
35
47
47
51
54
60
63
66
71
81
81
87
92
95
99
xi

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xii

/>CONTENTS

4

5

6

7

/>
19.
20.
21 .
22.

The Method of Variation of Parameters
Vibrations in Mechanical and Electrical Systems
Newton's Law of Gravitation and the Motion of the Planets
Higher Order Linear Equations. Coupled Harmonic
Oscillators
23. Operator Methods for Finding Particular Solutions
Appendix A. Euler
Appendix B . Newton

103
106

115

Qualitative Properties of Solutions

155
155
161

24. Oscillations and the Sturm Separation Theorem
25. The Sturm Comparison Theorem
Power Series Solutions and Special Functions

26.
27.
28.
29.
30.
31.
32.

Introduction. A Review of Power Series
Series Solutions of First Order Equations
Second Order Linear Equtions. Ordinary Points
Regular Singular Points
Regular Singular Points (Continued )
Gauss's Hypergeometric Equation
The Point at Infinity
Appendix A. Two Convergence Proofs
Appendix B . Hermite Polynomials and Quantum
Mechanics

Appendix C. Gauss
Appendix D . Chebyshev Polynomials and the Minimax
Property
Appendix E. Riemann's Equation

Fourier Series and Orthogonal Functions

33.
34.
35.
36.
37.
38.

The Fourier Coefficients
The Problem of Convergence
Even and Odd Functions. Cosine and Sine Series
Extension to Arbitrary Intervals
Orthogonal Functions
The Mean Convergence of Fourier Series
Appendix A. A Pointwise Convergence Theorem

Partial Differential Equations and Boundary
Value Problems

39.
40.
41.
42.
43 .


Introduction. Historical Remarks
Eigenvalues, Eigenfunctions, and the Vibrating String
The Heat Equation
The Dirichlet Problem for a Circle. Poisson's Integral
Sturm-Liouville Problems

122
128
136
146

165
165
172
176
184
192
199
204
208
211
221
230
237
246
246
257
265
272

277
285
293
298
298
302
311
317
323

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CONTENTS

Appendix A. The Existence of Eigenvalues and
Eigenfunctions
8

9

10

11

12

Some Special Functions of Mathematical Physics


44.
45.
46.
47.

Legendre Polynomials
Properties of Legendre Polynomials
Bessel Functions. The Gamma Function
Properties of Bessel functions
Appendix A. Legendre Polynomials and Potential Theory
Appendix B . Bessel Functions and the Vibrating
Membrane
Appendix C. Additional Properties of Bessel Functions

Laplace Transforms

48.
49.
50.
51.
52.
53.

Introduction
A Few Remarks on the Theory
Applications to Differential Equations
Derivatives and Integrals o f Laplace Transforms
Convolutions and Abel's Mechanical Problem
More about Convolutions. The Unit Step and Impulse
Functions

Appendix A. Laplace
Appendix B. Abel

Systems of First Order Equations

54.
55.
56.
57.

:xiii

/>
General Remarks on Systems
Linear Systems
Homogeneous Linear Systems with Constant Coefficients
Nonlinear Systems. Volterra's Prey- Predator Equations

Nonlinear Equations

58.
59.
60.
61.
62.
63.

Autonomous Systems. The Phase Plane and Its Phenomena
Types of Critical Points. Stability
Critical Points and Stability for Linear Systems

Stability by Liapunov's Direct Method
Simple Critical Points of Nonlinear Systems
Nonlinear Mechanics. Conservative Systems
64. Periodic Solutions. The Poincare- Bendixson Theorem
Appendix A. Poincare
Appendix B. Proof of Lienard's Theorem

The Calculus of Variations

65. Introduction. Some Typical Problems of the Subject
66. Euler's Differential Equation for an Extremal

33 1
335
335
342
348
358
365
371
377
381
381
385
390
394
399
405
412
413

417
417
421
427
434
440
440
446
455
465
471
480
486
494
497
502
502
505

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xiV

CONTENTS

/>
67. Isoperimetric problems
Appendix A. Lagrange

Appendix B . Hamilton's Principle and Its Implications

515
524
526

13

The Existence and Uniqueness of Solutions

538
538
543
552

14

Numerical Methods

68. The Method of Successive Approximations
69. Picard's Theorem
70. Systems. The Second Order Linear Equation
71.
72.
73.
74.
75.
76.

Introduction

The Method of Euler
Errors
An Improvement to Euler
Higher-Order Methods
Systems

Numerical Tables
Answers
Index

556
556
559
563
565
569
573
577
585
617

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PREFACE TO THE
SECOND EDITION

"As correct as a second edition"-so goes the idiom . I certainly hope so ,

and I also hope that anyone who detects an error will do me the kindness
of letting me know , so that repairs can be made . As Confucius said , "A
man who makes a mistake and doesn't correct it is making two
mistakes. "
I now understand why second editions of textbooks are always
longer than first editions: as with governments and their budgets , there is
always strong pressure from lobbyists to put things in, but rarely pressure
to take things out.
The main changes in this new edition are as follows: the number of
problems in the first part of the book has been more than doubled; there
are two new chapters , on Fourier Series and on Partial Differential
Equations ; sections on higher order linear equations and operator
methods have been added to Chapter 3; and further material on
convolutions and engineering applications has been added to the chapter
on Laplace Transforms.
Altogether , many different one-semester courses can be built on
various parts of this book by using the schematic outline of the chapters
given on page xxi . There is even enough material here for a two­
semester course , if the appendices are taken into account .
Finally , an entirely new chapter on Numerical Methods (Chapter
14) has been written especially for this edition by Major John S .
Robertson of the United States Military Academy . Major Robertson's
expertise in these matters is much greater than my own, and I am sure
that many users of this new edition will appreciate his contribution , as
I do.
McGraw-Hill and I would like to thank the following reviewers for
their many helpful comments and suggestions: D. R. Arterburn , New
XV

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XVi

PREFACE TO THE SECOND EDITION

/>
Mexico Tech ; Edward Beckenstein , St. John's University ; Harold Carda,
South Dakota School of Mines and Technology ; Wenxiong Chen,
University of Arizona; Jerald P. Dauer, University of Tennessee ;
Lester B . Fuller, Rochester Institute of Technology ; Juan Gatica,
University of Iowa; Richard H. Herman , The Pennsylvania State Univer­
sity; Roger H. Marty, Cleveland State University ; Jean-Pierre Meyer,
The Johns Hopkins University ; Krzysztof Ostaszewski, University of
Louisville ; James L. Rovnyak , University of Virginia; Alan Sharples,
New Mexico Tech ; Bernard Shiffman , The Johns Hopkins University ;
and Calvin H. Wilcox , University of Utah .

George F. Simmons

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PREFACE TO THE
FIRST EDITION

To be worthy of serious attention , a new textbook on an old subject

should embody a definite and reasonable point of view which is not
represented by books already in print. Such a point of view inevitably
reflects the experience , taste , and biases of the author, and should
therefore be clearly stated at the beginning so that those who disagree
can seek nourishment elsewhere . The structure and contents of this book
express my personal opinions in a variety of ways, as follows.
The place of dift'erential equations in mathematics. Analysis has been the
dominant branch of mathematics for 300 years , and differential equations
are the heart of analysis . This subject is the natural goal of elementary
calculus and the most important part of mathematics for understanding
the physical sciences. Also , in the deeper questions it generates , it is the
source of most of the ideas and theories which constitute higher analysis.
Power series, Fourier series, the gamma function and other special
functions , integral equations ,. existence theorems, the need for rigorous
justifications of many analytic processes-all these themes arise in our
work in their most natural context . And at a later stage they provide the
principal motivation behind complex analysis, the theory of Fourier series
and more general orthogonal expansions, Lebesgue integration , metric
spaces and Hilbert spaces, and a host of other beautiful topics in modern
mathematics. I would argue , for example , that one of the main ideas of
complex analysis is the liberation of power series from the confining
environment of the real number system ; and this motive is most clearly
felt by those who have tried to use real power series to solve differential
equations. In botany, it is obvious that no one can fully appreciate the
blossoms of flowering plants without a reasonable understanding of the
roots, stems , and leaves which nourish and support them . The same
principle is true in mathematics , but is often neglected or forgotten.

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PREFACE TO THE FI RST EDITION

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Fads are as common in mathematics as in any other human activity,
and it is always difficult to separate the enduring from the ephemeral in
the achievements of one's own time . At present there is a strong current
of abstraction flowing through our graduate schools of mathematics. This
current has scoured away many of the individual features of the
landscape and replaced them with the smooth , rounded boulders of
general theories. When taken in moderation , these general theories are
both useful and satisfying; but one unfortunate effect of their pre­
dominance is that if a student doesn't learn a little while he is an
undergraduate about such colorful and worthwhile topics as the wave
equation, Gauss's hypergeometric function , the gamma function , and the
basic problems of the calculus of variations-among many others-then
he is unlikely to do so later. The natural place for an informal
acquaintance with such ideas is a leisurely introductory course on
differential equations. Some of our current books on this subject remind
me of a sightseeing bus whose driver is so obsessed with speeding along
to meet a schedule that his passengers have little or no opportunity to
enjoy the scenery. Let us be late occasionally , and take greater pleasure
in the journey.
Applications. It is a truism that nothing is permanent except change ; and

the primary purpose of differential equations is to serve as a tool for the

study of change in the physical world . A general book on the subject
without a reasonable account of its scientific applications would therefore
be as futile and pointless as a treatise on eggs that did not mention their
reproductive purpose . This book is constructed so that each chapter
except the last has at least one major "payoff"-and often several-in the
form of a classic scientific problem which the methods of that chapter
render accessible. These applications include
The brachistochrone problem
The Einstein formulaE = mc 2
Newton's law of gravitation
The wave equation for the vibrating string
The harmonic oscillator in quantum mechanics
Potential theory
The wave equation for the vibrating membrane
The prey-predator equations
Nonlinear mechanics
Hamilton's principle
Abel's mechanical problem
I consider the mathematical treatment of these problems to be among the
chief glories of Western civilization , and I hope the reader will agree .

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PREFACE TO THE FI RST EDITION

XiX

of mathematical rigor. On the heights of pure mathematics,

any argument that purports to be a proof must be capable of withstanding
the severest criticisms of skeptical experts . This is one of the rules of the
game , and if you wish to play you must abide by the rules . But this is not
the only game in town .
There are some parts of mathematics-perhaps number theory and
abstract algebra-in which high standards of rigorous proof may be
appropriate at all levels. But in elementary differential equations a
narrow insistence on doctrinaire exactitude tends to squeeze the juice out
of the subject , so that only the dry husk remains. My main purpose in
this book is to help the student grasp the nature and significance of
differential equations ; and to this end , I much prefer being occasionally
imprecise but understandable to being completely accurate but incom­
prehensible . I am not at all interested in building a logically impeccable
mathematical structure , in which definitions, theorems, and rigorous
proofs are welded together into a formidable barrier which the reader is
challenged to penetrate .
In spite of these disclaimers , I do attempt a fairly rigorous
discussion from time to time , notably in Chapter 13 and Appendices A in
Chapters 5, 6 and 7, and B in Chapter 1 1 . I am not saying that the rest of
this book is nonrigorous , but only that it leans toward the activist school
of mathematics, whose primary aim is to develop methods for solving
scientific problems-in contrast to the contemplative school , which
analyzes and organizes the ideas and tools generated by the activists.
Some will think that a mathematical argument either is a proof or is
not a proof. In the context of elementary analysis I disagree , and believe
instead that the proper role of a proof is to carry reasonable conviction to
one's intended audience . It seems to me that mathematical rigor is like
clothing: in its style it ought to suit the occasion , and it diminishes
comfort and restricts freedom of movement if it is either too loose or too
tight.


The problem

History and biography. There is an old Armenian saying, "He who lacks a
sense of the past is condemned to live in the narrow darkness of his own
generation." Mathematics without history is mathematics stripped of its
greatness: for, like the other arts--and mathematics is one of the
supreme arts of civilization-it derives its grandeur from the fact of being
a human creation .
I n a n age increasingly dominated b y mass culture and bureaucratic
impersonality , I take great pleasure in knowing that the vital ideas of
mathematics were not printed out by a computer or voted through by a
committee , but instead were created by the solitary labor and individual
genius of a few remarkable men . The many biographical notes in this
book re flect my desire to convey something of the achievements and
personal qualities of these astonishing human beings. Most of the longer

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XX

/>
PREFACE TO THE FI RST EDITION

notes are placed in the appendices , but each is linked directly to a specific
contribution discussed in the text. These notes have as their subjects all
but a few of the greatest mathematicians of the past three centuries:
Fermat , Newton , the Bernoullis, Euler, Lagrange , Laplace , Fourier,

Gauss , Abel, Poisson , Dirichlet , Hamilton, Liouville , Chebyshev , Herm­
ite , Riemann , Minkowski , and Poincare. As T. S. Eliot wrote in one of
his essays, "Someone said : 'The dead writers are remote from us because
we know so much more than they did . ' Precisely, and they are that which
we know. "
History and biography are very complex , and I am painfully aware
that scarcely anything in my notes is actually quite as simple as it may
appear . I must also apologize for the many excessively brief allusions to
mathematical ideas most student readers have not yet encountered . But
with the aid of a good library, sufficiently interested students should be
able to unravel most of them for themselves. At the very least , such
efforts may help to impart a feeling for the immense diversity of classical
mathematics-an aspect of the subject that is almost invisible in the
average undergraduate curriculum .

George F. Simmons

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SUGGESTIONS FOR THE
INSTRUCTOR

The following diagram gives the logical dependence of the chapters and
suggests a variety of ways this book can be used, depending on the
purposes of the course , the tastes of the instructor, and the backgrounds
and needs of the students.
I.


The Nature or
Difl'ercntial
Equations Separable
Equations

13

Existence and
Uniqueness

14.

Numerical
Methods

Theorems

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The scientist does not study nature because it is useful; he studies it because
he delights in it, and he delights in it because it is beautiful. If nature were

not beautiful, it would not be worth knowing, and if nature were not worth
knowing, life would not be worth living. Of course I do not here speak of
that beauty that strikes the senses, the beauty of qualities and appearances;
not that I undervalue such beauty, far from it, but it has nothing to do with
science; I mean that profounder beauty which comes from the harmonious
order of the parts, and which a pure intelligence can grasp .
-Henri Poincare

As a mathematical discipline travels far from its empirical source, or still
more, if it is a second or third generation only indirectly inspired by ideas
coming from "reality," it is beset with very grave dangers. It becomes
more and more purely aestheticizing, more and more purely l art pour
I'art. This need not be bad, if the field is surrounded by correlated
subjects, which still have closer empirical connections, or if the discipline
is under the influence of men with an exceptionally well-developed taste.
But there is a grave danger that the subject will develop along the line of
least resistance, that the stream, so far from its source, will separate into a
multitude of insignificant branches, and that the discipline will become a
disorganized mass of details and complexities . In other words, at a great
distance from its empirical source, or after much "abstract" inbreeding, a
mathematical subject is in danger of degeneration.
'

-John von Neumann

Just as deduction should be supplemented by intuition, so the impulse to
progressive generalization must be tempered and balanced by respect and
love for colorful detail. The individual problem should not be degraded to
the rank of special illustration of lofty general theories. In fact, general
theories emerge from consideration of the specific, and they are meaning­

less if they do not serve to clarify and order the more particularized
substance below. The interplay between generality and individuality,
deduction and construction, logic and imagination-this is the profound
essence of live mathematics. Any one or another of these aspects of
mathematics can be at the center of a given achievement. In a far-reaching
development all of them will be involved. Generally speaking, such a
development will start from the "concrete" ground, then discard ballast by
abstraction and rise to the lofty layers of thin air where navigation and
observation are easy; after this flight comes the crucial test of landing and
reaching specific goals in the newly surveyed low plains of individual
"reality." In brief, the flight into abstract generality must start from and
return to the concrete and specific.
-Richard Courant

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DIFFERENTIAL EQUATIONS
WITH APPLICATIONS AND HISTORICAL NOTES

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CHAPTER

1
THE
NATURE OF
DIFFERENTIAL
EQUATIONS.
SEPARABLE
EQUATIONS

1

INTRODUCfiON

An equation involving one dependent variable and its derivatives with
respect to one or more independent variables is called a differential
equation . Many of the general laws of nature-in physics , chemistry ,
biology and astronomy-find their most natural expression in the
langua ge of differential equations. Applications also abound in mathe­
matics itself, especially in geometry, and in engineering, economics, and
many other fields of applied science.
It is easy to understand the reason behind this broad utility of
differential equations. The reader will recall that if y = f(x) is a given

function, then its derivative dy/dx can be interpreted as the rate of
change of y with respect to x. In any natural process , the variables
involved and their rates of change are connected with one another by
means of the basic scientific principles that govern the process. When this
connection is expresse d in mathem atical symbols, the result is often a
differential eq uat ion .
The following ex ample may illuminate these remarks. According to
Newton's second law of motion, the acceleration a of a body of mass m is
proportional to the total force F acting on it , with 1/m as the constant of
,

1

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