Differential equations : theory, technique, and practice

Titles in the Walter Rudin Student Series in Advanced Mathematics

Bona, Miklos

Introduction to Enumerative Combinatorics

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Transition to Higher Mathematics: Structure and Proof

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Functional Analysis, 2nd Edition

Rudin, Walter

Principles of Mathematical Analysis, 3rd Edition

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Real and Complex Analysis, 3rd Edition

Simmons, George F. and Steven G. Krantz

Differential Equations: Theory, Technique, and Practice

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Differential

Equations

Theory, Technique,

and Practice

George F. Simmons

Colorado College

and

Steven G. Krantz

Washington University

in St. Louis

!R Higher Education

Boston Burr Ridge, IL Dubuque, IA Madison, WI New York San Francisco St. Louis

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DIFFERENTIAL EQUATIONS: THEORY, TECHNIQUE, AN D PRACTICE

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Library of Congress Cataloging-in-Publication Data

Simmons, George Finlay, 1925-.

Differential equations : theory, technique, and practice/George F. Simmons, Steven G. Krantz. - !st ed.

p. cm.

Includes bibliographical references and index.

ISBN 978-0-07-286315-4-ISBN 0-07-286315-3 (acid-free paper)

1. Differential equations-Textbooks. I. Krantz, Steven G. (Steven George), 1951-. II. Title.

QA371.S465 2007

515 .35-<lc22

www.mhhe.com

2005051118

CIP

PREFACE VIII

CHAPTER 1 WHAT IS A DIFFERENTIAL EQUATION? 1

1.1 Introductory Remarks 2

1.2 The Nature of Solutions 4

1.3 Separable Equations 10

1.4 First-Order Linear Equations 13

1.5 Exact Equations 17

1.6 Orthogonal Trajectories and Families of Curves 22

1.7 Homogeneous Equations 26

1.8 Integrating Factors 29

1.9 Reduction of Order 33

1.9.l Dependent Variable Missing 33

1.9.2 Independent Variable Missing 35

1.10 The Hanging Chain and Pursuit Curves 38

1.10.1 The Hanging Chain 38

1.10.2 Pursuit Curves 42

1.11 Electrical Circuits 45

Anatomy of an Application: The Design of a

Dialysis Machine 49

Problems for Review and Discovery 53

CHAPTER 2 SECOND-ORDER LINEAR EQUATIONS 57

2.1 Second-Order Linear Equations with Constant

Coefficients 58

2.2 The Method of Undetermined Coefficients 63

2.3 The Method of Variation of Parameters 67

2.4 The Use of a Known Solution to Find Another 71

2.5 Vibrations and Oscillations 75

2.5.1 Undamped Simple Harmonic Motion 75

2.5.2 Damped Vibrations 77

2.5.3 Forced Vibrations 80

2.5.4 A Few Remarks about Electricity 82

iii

iv Table of Contents

2.6 Newton's Law of Gravitation and Kepler's Laws 84

2.6.l Kepler's Second Law 86

2.6.2 Kepler's First Law 87

2.6.3 Kepler's Third Law 89

2.7 Higher Order Linear Equations, Coupled

Harmonic Oscillators 93

Historical Note: Euler 99

Anatomy of an Application: Bessel

Functions and the Vibrating Membrane 101

Problems for Review and Discovery 105

CHAPTER 3 QUALITATIVE PROPERTIES AND·

THEORETICAL ASPECTS 109

3.1 Review of Linear Algebra 110

3.1. l Vector Spaces 110

3.1.2 The Concept of Linear Independence 111

3.1.3 Bases 113

3.1.4 Inner Product Spaces 114

3.1.5 Linear Transformations and Matrices 115

3.1.6 Eigenvalues and Eigenvectors 117

3.2 A Bit of Theory 119

3.3 Picard's Existence and Uniqueness Theorem 125

3.3.l The Form of a Differential Equation 125

3.3.2 Picard's Iteration Technique 126

3.3.3 Some Illustrative Examples 127

3.3.4 Estimation of the Picard Iterates 129

3.4 Oscillations and the Sturm Separation Theorem 130

3.5 The Sturm Comparison Theorem 138

Anatomy of an Application: The Green's

Function 142

Problems for Review and Discovery 146

CHAPTER 4 POWER SERIES SOLUTIONS AND

SPECIAL FUNCTIONS 149

4.1 Introduction and Review of Power Series 150

4.1.l Review of Power Series 150

4.2 Series Solutions of First-Order Differential

Equations 159

CHAPTER 5

CHAPTER 6

Table of Contents v

4.3 Second-Order Linear Equations: Ordinary Points 164

4.4 Regular Singular Points 171

4.5 More on Regular Singular Points 177

4.6 Gauss's Hypergeometric Equation 184

Historical Note: Gauss 189

Historical Note: Abel 190

Anatomy of an Application: Steady-State

Temperature in a Ball 192

Problems for Review and Discovery 194

FOURIER SERIES: BASIC CONCEPTS 197

5.1 Fourier Coefficients 198

5.2 Some Remarks about Convergence 207

5.3 Even and Odd Functions: Cosine and Sine Series 211

5.4 Fourier Series on Arbitrary Intervals 218

5.5 Orthogonal Functions 221

Historical Note: Riemann 225

Anatomy of an Application: Introduction to

the Fourier Transform 227

Problems for Review and Discovery

PARTIAL DIFFERENTIAL EQUATIONS

AND BOUNDARY VALUE PROBLEMS

236

239

6.1 Introduction and Historical Remarks 240

6.2 Eigenvalues, Eigenfunctions, and the Vibrating

String 243

6.2.1 Boundary Value Problems 243

6.2.2 Derivation of the Wave Equation

6.2.3 Solution of the Wave Equation

6.3 The Heat Equation

6.4 The Dirichlet Problem for a Disc

6.4.1 The Poisson Integral

6.5 Sturm-Liouville Problems

Historical Note: Fourier

Historical Note: Dirichlet

Anatomy of an Application: Some Ideas

from Quantum Mechanics

Problems for Review and Discovery

244

246

251

256

259

262

267

268

270

273

vi Table of Contents

CHAPTER 7 LAPLACE TRANSFORMS 277

7.1 Introduction 278

7.2 Applications to Differential Equations 280

7.3 Derivatives and Integrals of Laplace Transforms 285

7.4 Convolutions 291

7.4.1 Abel's Mechanical Problem 293

7.5 The Unit Step and Impulse Functions 298

Historical Note: Laplace 305

Anatomy of an Application: Flow Initiated by

an Impulsively Started Flat Plate · 306

Problems for Review and Discovery 309

CHAPTER 8 THE CALCULUS OF VARIATIONS 315

8.1 Introductory Remarks 316

8.2 Euler's Equation 319

8.3 Isoperimetric Problems and the Like 327

8.3.1 Lagrange Multipliers 328

8.3.2 Integral Side Conditions 329

8.3.3 Finite Side Conditions 333

Historical Note: Newton 338

Anatomy of an Application: Hamilton's

Principle and its Implications 340

Problems for Review and Discovery 344

CHAPTER 9 NUMERICAL METHODS 347

9.1 Introductory Remarks 348

9.2 The Method of Euler 349

9.3 The Error Term 353

9.4 An Improved Euler Method 357

9.5 The Runge-Kutta Method 360

Anatomy of an Application: A Constant

Perturbation Method for Linear,

Second-Order Equations 365

Problems for Review and Discovery 368

CHAPTER 10 SYSTEMS OF FIRST-ORDER

EQUATIONS 371

10.l Introductory Remarks 372

10.2 Linear Systems 374

CHAPTER 11

CHAPTER 12

Table of Contents

10.3 Homogeneous Linear Systems with Constant

vii

Coefficients 382

10.4 Nonlinear Systems: Volterra's Predator-Prey

Equations 389

Anatomy of an Application: Solution of

Systems with Matrices and Exponentials 395

Problems for Review and Discovery 400

THE NONLINEAR THEORY 403

11.1 Some Motivating Examples 404

11.2 Specializing Down 404

11.3 Types of Critical Points: Stability 409

11.4 Critical Points and Stability for Linear Systems 417

11.5 Stability by Liapunov's Direct Method 427

11.6 Simple Critical Points of Nonlinear Systems 432

11.7 Nonlinear Mechanics: Conservative Systems 439

11.8 Periodic Solutions: The Poincare-Bendixson

Theorem

Historical Note: Poincare

Anatomy of an Application: Mechanical

Analysis of a Block on a Spring

Problems for Review and Discovery

DYNAMICAL SYSTEMS

12.1 Flows

12.1.1 Dynamical Systems

12.1.2 Stable and Unstable Fixed Points

12.1.3 Linear Dynamics in the Plane

12.2 Some Ideas from Topology

12.2.1 Open and Closed Sets

12.2.2 The Idea of Connectedness

444

452

454

457

461

462

464

466

468

475

475

476

12.2.3 Closed Curves in the Plane 478

12.3 Planar Autonomous Systems 480

12.3.1 Ingredients of the Proof of Poincare-Bendixson 480

Anatomy of an Application: Lagrange's

Equations 489

Problems for Review and Discovery

BIB LIOG RAP HY

ANSWERS TO ODD-NUMBERED EXERCISES

INDEX

493

495

497

525

Differential equations is one of the oldest subjects in modem mathematics. It was not Jong after

Newton and Leibniz invented calculus that Bernoulli and Euler and others began to consider the

heat equation and the wave equation of mathematical physics. Newton himself solved differential

equations both in the study of planetary motion and also in his consideration of optics.

Today differential equations is the centerpiece of much of engineering, of physics, of signif￾icant parts of the life sciences, and in many areas of mathematical modeling. The audience for a

sophomore course in ordinary differential equations is substantial-second only perhaps to that

for calculus. There is a need for a definitive text that both describes classical ideas and provides

an entree to the newer ones. Such a text should pay careful attention to advan_ced topics like the

Laplace transform, Sturm-Liouville theory, and boundary value problems (on the traditional side)

but should also pay due homage to nonlinear theory, to dynamics, to modeling, and to computing

(on the modem side).

George Simmons's fine text is a traditional book written in the classical style. It provides

a cogent and accessible introduction to all the traditional topics. It is a pleasure to have this

opportunity to bring this text up to date and to add some more timely material. We have streamlined

some of the exposition and augmented other parts. There is now computer work based not only on

number crunching but also on computer algebra systems such as Maple, Mathematica, and

.MATLAB. Certainly a study of flows and vector fields, and of the beautiful Poincare-Bendixson

theory built thereon, is essential for any modem treatment. One can introduce some of the modem

ideas from the theory of dynamics to obtain qualitative information about nonlinear differential

equations and systems.

And all of the above is a basis for modeling. Modeling is what brings the subject to life

and makes the ideas real for the students. Differential equations can model real-life questions,

and computer calculations and graphics can then provide real-life answers. The symbiosis of the

synthetic and the calculational provides a rich educational experience for students, and it prepares

them for more concrete, applied work in future courses. The new Anatomy of an Application

sections in this edition showcase some rich applications from engineering, physics, and applied

science.

There are a number of good ordinary differential equations books available today. Popu￾lar standards include Boyce & DiPrima; Nagle, Saff, & Snider; Edwards & Penney; Derrick

& Grossman; and Polking, Boggess & Arnold. Books for a more specialized audience include

Amol'd; Hubbard & Hubbard; Borrelli & Coleman; and Blanchard, Devaney, & Hall. Classical

books, still in use at some schools, include Coddington & Levinson and Birkhoff & Rota. Each

of these books has some strengths, but not the combination of features that we have planned for

Simmons & Krantz. None has the crystal clear and elegant quality of writing for which George

Simmons is so well known. Steven G. Krantz is also a mathematical writer of some repute (50 books

and 140 papers), and can sustain the model set by Simmons in this new edition. No book will

have the well-developed treatment of modeling and computing (done in a manner so that these

two activities speak to each other) that will be rendered in Simmons & Krantz. None will have the

quality of exercises.

We look forward to setting a new standard for the modem textbook on ordinary differential

equations, a standard to which other texts may aspire. This will be a book that students read, and

internalize, and in the end apply to other subjects and disciplines. It will Jay the foundation for

Preface ix

future studies in analytical thinking. It will be a touchstone for in-depth studies of growth and

change.

Key Features

• Anatomy of An Application - Occurring at the end of each chapter, these in-depth exami￾nations of particular applications of ordinary differential equations motivate students to use

critical thinking skills to solve practical problems in engineering, physics, and the sciences.

After the application is introduced in context, the key concepts and procedures needed to

model its associated problems are presented and discussed in detail.

• Exercises - The text contains a wide variety of section-level exercises covering varying

levels of difficulty. Hints are given when appropriate to assist students with difficult prob￾lems and crucial concepts. Special technology exercises are included in nearly every section

which harness the power of computer algebra systems such as Maple, Mathematica, and

MATLAB for solving ordinary differential equations. Answers to the odd-numbered exercises

in the text are included in the Answers to Odd-Numbered Exercises at the back of the book.

• Problems for Review and Discovery - Each chapter is concluded with three sets of review

exercises. Drill Exercises test students' basic understanding of key concepts from the chapter.

Challenge Problems take that review a step further by presenting students with more complex

problems requiring a greater degree of critical thinking. And Problems for Discussion and

Exploration offer students open-ended opportunities to explore topics from the chapter and

develop their intuition and command of the material.

• Historical Notes - These biographies, occurring at the end of chapters, offer fascinating in￾sight into the lives and accomplishments of some of the great mathematicians who contributed

to the development of differential equations. A longtime hallmark of George Simmons' writ￾ings, the Historical Notes show how mathematics is at its heart a human endeavor developed

to meet human needs.

• Math Nuggets - These brief asides, appearing throughout the text, offer quick historical

context and interesting anecdotes tied to the specific topic under discussion. They serve to

underscore the human element behind the development of ordinary differential equ<1tions in

shorter and more context-sensitive form than the end-of-chapter Historical Notes.

Supplements

Student's Solutions Manual, by Donald Hartig (ISBN-JO: 0-07-286316-1, ISBN-13: 978-0-07-

286316-1) - Contains complete worked solutions to odd-numbered exercises from the text.

Instructor's Solutions Manual, by Donald Hartig (ISBN-10: 0-07-323091-X, ISBN-13: 978-0-

07-323091-7) - Contains complete worked solutions to even-numbered exercises from the text.

Companion Website, http://www.mhhe.com/simmons - Contains free online resources for stu￾dents and instructors to accompany the text. The website features online technology manuals for

computer algebra systems such as Maple and Mathematica. These technology manuals give a

general overview of these systems and how to use them to solve and explore ordinary differen￾tial equations, and provide additional problems and worksheets for further practice with these

computational tools.

x Preface

Acknowledgements

We would like to thank the following individuals who reviewed the manuscript and provided

valuable suggestions for improvement:

Yuri Antipov, Louisiana State University

Dieter Armbruster, Arizona State University

Vitaly Bergelson, Ohio State University

James Ward Brown, University of Michigan - Dearborn

Nguyen Cac, University of Iowa

Benito Chen, University of Wyoming

Goong Chen, Texas A&M University

Ben Cox, College of Charleston

Richard Crew, University of Florida

Moses Glasner, Pennsylvania State University

David Grant, University of Colorado

Johnny Henderson, Baylor University

Michael Kirby, Colorado State University

Przemo Kranz, University of Mississippi

Melvin Lax, California State University - Long Beach

William Margulies, California State University - Long Beach

James Okon, California State University - San Bernardino

William Paulsen, Arkansas State University

Jonathan Rosenberg, University of Maryland

Jairo Santanilla, University of New Orleans

Michael Shearer, North Carolina State University

Jie Shen, Purdue University

Marshall Slemrod, University of Wisconsin

P.K. Subramanian, California State University - Los Angeles

Kirk Tolman, Brigham Young University

Xiaoming Wang, Florida State University

Steve Zelditch, Johns Hopkins University

Zhengfang Zhou, Michigan State University

Special thanks go to Steven Boettcher, who prepared the answer key; Donald Hartig, who

prepared the two solutions manuals; and Daniel Zwillinger, who checked the complete text and

exercises for accuracy.

The previous incarnation of this classic text was written by George F. Simmons. His book

has served as an inspiration to several generations of differential equations students. It has been

a pleasure to prepare this new version for a new body of students. George Simmons has played a

proactive role at every stage of the writing process, contributing many ideas, edits, and corrections.

His wisdom pervades the entire text.

Steven G. Krantz

For Hope and Nancy

my wife and daughter

who still make it all worthwhile

For Randi and Hypatia

my wife and daughter

who know why I dedicate my books to them

CHAPTER 1

-·

--

--

-- -

--

--

-W �t Is a Differential

• The concept of a differential equation

• Characteristics of a solution

• Finding a solution

• Separable equations

• First-order linear equations

• Exact equations

• Orthogonal trajectories

Equation?

2 Chapter 1 What Is a Differential Equation? ,... INTRODUCTORY REMARKS

A differential equation is an equation relating some function f to one or more of its

derivatives. An example is

d

2f df -2

(x) + 2x-(x) + f2

(x) = sin x. dx dx (1.1)

Observe that this particular equation involves a function f together with its first and

second derivatives. Any given differential equation may or may not involve f or any

particular derivative off. But, for an equation to be a differential equation, at least some

derivative off must appear. The objective in solving an equation like Equation (1.1) is

to find the function f. Thus we already perceive a fundamental new paradigm: When we

solve an algebraic equation, we seek a number or perhaps a collection of numbers; but

when we solve a differential equation we seek one or more functions.

Many of the laws of nature-in physics, in chemistry, in biology, in engineering, and

in astronomy-find their most natural expression in the language of differential equa￾tions. Put in other words, differential equations are the language of nature. Applications

of differential equations also abound in mathematics itself, especially in geometry and

harmonic analysis and modeling. Differential equations occur in economics and systems

science and other fields of mathematical science.

It is not difficult to perceive why differential equations arise so readily in the sciences.

If y = f(x) is a given function, then the derivativedf /dx can be interpreted as the rate of

change off with respect to x. In any process of nature, the variables involved are related

to their rates of change by the basic scientific principles that govern the process-that

is, by the laws of nature. When this relationship is expressed in mathematical notation,

the result is usually a differential equation.

Certainly Newton's law of universal gravitation, Maxwell's field equations, the

motions of the planets, and the refraction of light are important examples which can be

expressed using differential equations. Much of our understanding of nature comes from

our ability to solve differential equations. The purpose of this book is to introduce you

to some of these techniques.

The following example will illustrate some of these ideas. 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 the body. The standard expression of this relationship is

F =m · a . (1.2)

Suppose in particular that we are analyzing a falling body. Express the height of the

body from the surface of the Earth as y(t) feet at time t. The only force acting on the

body is that due to gravity. If g is the acceleration due to gravity (about -32 ft/sec2 near

the surface of the Earth) then the force exerted on the body ism · g. And of course the

acceleration is d2y/dt2• Thus Newton's law, Equation (1.2), becomes

(1.3)

or

Section 1.1 Introductory Remarks

d2y

g = dt2 •

3

We may make the problem a little more interesting by supposing that air exerts a

resisting force proportional to the velocity. If the constant of proportionality is k, then

the total force acting on the body is mg - k · (dy/dt). Then Equation (1.3) becomes

dy d2y

m · g - k · - = m · -. (1.4)

dt dt2

Equations (1.3) and (1.4) express the essential attributes of this physical system.

A few additional examples of differential equations are

2 d2y dy (1 - x ) dx2 -2x

dx + p(p + l)y = 0;

d2y dy

x2- +x- + {x2 - p2) y = O; dx2 dx

d2y

dx2 +xy = O;

(1-x2)y" -xy' + p2y = O;

y" - 2xy' + 2py = 0;

-dy =k·y; dx

d3y

3 + (dy ) 2

= y 3 + sinx. dx dx

(1.5)

(1.6)

(1.7)

(1.8)

(1.9)

(1.10)

(1.11)

Equations (l.5)-(1.9) are called Legendre's equation, Bessel's equation, Airy's equation,

Chebyshev's equation, and Hermite's equation, respectively. Each has a vast literature

and a history reaching back hundreds of years. We shall touch on each of these equations

later in the book. Equation ( 1.10) is the equation of exponential decay (or of biological

growth).

Adrien Marie Legendre (1752-1833) invented Legendre polynomials (the contri￾bution for which he is best remembered) in the context of gravitational attraction

of ellipsoids. Legendre was a fine French mathematician who suffered the misfor￾tune of seeing most of his best work-in elliptic integrals, number theory, and the

method of least squares-superseded by the achievements of younger and abler

men. For instance, he devoted 40 years to the study of elliptic integrals, and his