An Introduction to Differential Equations: With Difference Equations, Fourier Series, and Partial Differential Equations

495 Pages·1982·5.64 MB·English

by  N. Finizio

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with Difference Equations. Fourier Analysis, and Partial Differential Equations
1 ? 1 C/ L A ID A S
1= I ICI
An Introduction to
Differential Equations
with Difference Equations, Fourier Series,
and Partial Differential Equations
N. Finizio and G. Ladas
University of Rhode Island
Wadsworth Publishing Company
Belmont, California
A division of Wadsworth, Inc.
ABOUT THE COVER: "Dirichlet Problem"
(Miles Color Art A25) is the work of Professor
E. P. Miles, Jr., and associates of Florida
State University using an InteColor 80501
computer. Programming by Eric Chamberlain
and photograph by John Owen. The function
graphed is the discrete limiting position for the
solution by relaxation of the heat distribution
in an insulated rectangular plate with fixed
temperature at boundary positions. This is an
end-position photograph following intermediate
positions displayed as the solution converges
from an assumed initial average temperature to
the ultimate (harmonic function) steady-state
temperature induced by the constantly
maintained boundary conditions.
Mathematics Editor: RICHARD JONES
Signing Representative: RICHARD GIOGEV
® 1982 by Wadsworth, Inc.
All rights reserved. No part of this book may be
reproduced, stored in a retrieval system, or transcribed, in
any form or by any means, electronic, mechanical,
photocopying, recording, or otherwise, without the prior
written permission of the publisher, Wadsworth Publishing
Company, Belmont, California 94002, a division of
Wadsworth, Inc.
ISBN 0-534-00960-3
Printed in the United States of America
5 6 7 8 9 10- 96 95 94 93 92 91
Ubrary of Congress Cataloging in Publication Data
Finizio. N.
An introduction to ordinary differential equations,
with difference equations. Fourier series, and
partial differential equations.
Includes index.
1. Differential equations. 1. Ladas, G.
11. Title.
QA372.F55 515.3'52 81-1971
ISBN 0-534-00960-3 AACR2
Preface
This book is designed for an introductory, one-semester or one-year course
in differential equations, both ordinary and partial. Its prerequisite is ele-
mentary calculus.
Perusal of the table of contents and the list of applications shows that the
book contains the theory, techniques, and applications covered in the tradi-
tional introductory courses in differential equations. A major feature of this
text is the quantity and variety of applications of current interest in physical,
biological, and social sciences. We have furnished a wealth of applications
from such diverse fields as astronomy, bioengineering, biology, botany, chem-
istry, ecology, economics, electric circuits, finance, geometry, mechanics,
medicine, meteorology, pharmacology, physics, psychology, seismology, so-
ciology, and statistics.
Our experience gained in teaching differential equations at the elementary,
intermediate, and graduate levels at the University of Rhode Island convinced
us of the need for a book at the elementary level which emphasizes to the
students the relevance of the various equations to which they are exposed in
the course. That is to say, that the various types of differential equations
encountered are not merely the product of some mathematician's imagination
but rather that the equations occur in the course of scientific investigations
of real-world phenomena.
The goal of this book, then, is to make elementary differential equations
more useful, more meaningful, and more exciting to the student. To accomplish
this, we strive to demonstrate that differential equations are very much "alive"
in present-day applications. This approach has indeed had a satisfying effect
in the courses we have taught recently.
During the preparation and class testing of this text we continuously kept
in mind both the student and the teacher. We have tried to make the pre-
sentation direct, yet informal. Definitions and theorems are stated precisely
and rigorously, but theory and rigor have been minimized in favor of com-
prehension of technique. The general approach is to use a larger number of
routine examples to illustrate the new concepts, definitions, methods of so-
lution, and theorems. Thus, it is intended that the material will be easily
accessible to the student. Hopefully the presence of modern applications in
addition to the traditional applications of geometry, physics, and chemistry
will be refreshing to the teacher.
Numerous routine exercises in each section will help to test and strengthen
the student's understanding of the new methods under discussion. There are
over 1600 exercises in the text with answers to odd-numbered exercises pro-
vided. Some thought-provoking exercises from The American Mathematical
X
Monthly, Mathematics Magazine, and The William Lowell Putnam Mathe-
matics Competition are inserted in many sections, with references to the
source. These should challenge the students and help to train them in searching
the literature. Review exercises appear at the end of every chapter. These
exercises should serve to help the student review the material presented in
the chapter. Some of the review exercises are problems that have been taken
directly from physics and engineering textbooks. The inclusion of such prob-
lems should further emphasize that differential equations are very much pres-
ent in applications and that the student is quite apt to encounter them in areas
other than mathematics.
Every type of differential equation studied and every method presented is
illustrated by real-life applications which are incorporated in the same section
(or chapter) with the specific equation or method. Thus, the student will see
immediately the importance of each type of differential equation that they
learn how to solve. We feel that these "modern" applications, even if the
student only glances at some of. them, will help to stimulate interest and
enthusiasm toward the subject of differential equations specifically and math-
ematics in general.
Many of the applications are integrated into the main development of ideas,
thus blending theory, technique, and application. Frequently, the mathematical
model underlying the application is developed in great detail. It would be
impossible in a text of this nature to have such development for every ap-
plication cited. Therefore, some of the models are only sketched, and in some
applications the model alone is presented. In practically all cases, references
are given for the source of the model. Additionally, a large number of appli-
cations appear in the exercises; these applications are also suitably referenced.
Consequently, applications are widespread throughout the book, and although
they vary in depth and difficulty, they should be diverse and interesting enough
to whet the appetite of every reader. As a general statement, every application
that appears, even those with little or no detail, is intended to illustrate the
relevance of differential equations outside of their intrinsic value as mathe-
matical topics. It is intended that the instructor will probably present only a
few of the applications, while the rest can demonstrate to the reader the
relevance of differential equations in real-life situations.
The first eight chapters of this book are reproduced from our text Ordinary
Differential Equations with Modern Applications, Second Edition, Wadsworth
Publishing Co., 1981.
The additional chapters 9, 10, and 11 treat difference equations, Fourier
series, and partial differential equations, respectively. Each of these chapters
provides a thorough introduction to its respective topic. We feel that the
chapters on difference equations and partial differential equations are more
extensive than one usually finds at this level. Our purpose for including these
topics is to allow more course options for users of this book.
We are grateful to Katherine MacDougall, who so skillfully typed the man-
uscript of this text.
A special word of gratitude goes to Professors Gerald Bradley, John Had-
dock, Thomas Hallam, Ken Kalmanson, Gordon McLeod, and David Wend,
Preface xl
who painstakingly reviewed portions of this book and offered numerous val-
uable suggestions for its improvement.
Thanks are also due to Dr. Clement McCalla, Dr. Lynnell Stern, and to
our students Carl Bender, Thomas Buonanno, Michael Fascitelli, and espe-
cially Neal Jamnik, Brian McCartin, and Nagaraj Rao who proofread parts
of the material and doublechecked the solutions to some of the exercises.
Special thanks are due to Richard Jones, the Mathematics Editor of Wads-
worth Publishing Company for his continuous support, advice, and active
interest in the development of this project.
N. Finizio
G. Ladas
Contents
1 ELEMENTARY METHODS-FIRST-ORDER DIFFERENTIAL EQUATIONS
1.1 Introduction and Definitions 1
1.1.1 Applications 3
1.2 Existence and Uniqueness 11
1.3 Variables Separable 17
1.3.1 Applications 20
1.4 First-Order Linear Differential Equations 30
1.4.1 Applications 33
1.5 Exact Differential Equations 41
1.5.1 Application 46
1.6 Homogeneous Equations 51
1.6.1 Application 53
1.7 Equations Reducible to First Order 55
1.7.1 Application 57
Review Exercises 58
2
LINEAR DIFFERENTIAL EQUATIONS
2.1 Introduction and Definitions 65
2.1.1 Applications 67
2.2 Linear Independence and Wronskians 73
2.3 Existence and Uniqueness of Solutions 81
2.4 Homogeneous Differential Equations with Constant Coefficients-
The Characteristic Equation 87
2.5 Homogeneous Differential Equations with Constant Coefficients-
The General Solution 92
2.5.1 Application 97
2.6 Homogeneous Equations with Variable Coefficients-
Overview 102
2.7 Euler Differential Equation 103
2.8 Reduction of Order 108
2.8.1 Applications 112
2.9 Solutions of Linear Homogeneous Differential Equations
by the Method of Taylor Series 116
Contents
xiv
2.10 Nonhomogeneous Differential Equations 120
2.11 The Method of Undetermined Coefficients 124
2.11.1 Applications 129
2.12 Variation of Parameters 135
Review Exercises 141
3
LINEAR SYSTEMS
3.1 Introduction and Basic Theory 147
3.11 Applications 153
3.2 The Method of Elimination 162
3.2.1 Applications 165
3.3 The Matrix Method 170
3.3.1 Nonhomogeneous Systems-Variation of Parameters 181
3.3.2 Applications 183
Review Exercises 187
4
THE LAPLACE TRANSFORM
4.1 Introduction 190
4.2 The Laplace Transform and Its Properties 190
4.3 The Laplace Transform Applied to Differential Equations
and Systems 199
4.4 The Unit Step Function 206
4.5 The Unit Impulse Function 210
4.6 Applications 214
Review Exercises 221
5
SERIES SOLUTIONS OF SECOND-ORDER LINEAR EQUATIONS
5.1 Introduction 224
5.2 Review of Power Series 225
5.3 Ordinary Points and Singular Points 229
5.4 Power-Series Solutions about an Ordinary Point 232
5.4.1 Applications 238
5.5 Series Solutions about a Regular Singular Point 246
5.5.1 Applications 258
Review Exercises 266
Contents xv
6 BOUNDARY VALUE PROBLEMS
61 Introduction and Solution of Boundary Value Problems 269
61.1 Applications 273
6.2 Eigenvalues and Eigenfunctions 276
6.2.1 Application 281
Review Exercises 285
7
NUMERICAL SOLUTIONS OF DIFFERENTIAL EQUATIONS
7.1 Introduction 286
7.2 Euler Method 288
7.3 Taylor-Series Method 293
7.4 Runge-Kutta Methods 298
7.5 Systems of First-Order Differential Equations 304
7.6 Applications 308
Review Exercises 312
8
NONLINEAR DIFFERENTIAL EQUATIONS AND SYSTEMS
8.1 Introduction 315
8.2 Existence and Uniqueness Theorems 315
8.3 Solutions and Trajectories of Autonomous Systems 317
8.4 Stability of Critical Points of Autonomous Systems 321
85 Phase Portraits of Autonomous Systems 327
8.6 Applications 337
Review Exercises 346
9
DIFFERENCE EQUATIONS
9.1 Introduction and Definitions 348
9.1.1 Applications 350
9.2 Existence and Uniqueness of Solutions 354
9.3 Linear Independence and the General Solution 358
9.4 Homogeneous Equations with Constant Coefficients 366
9.5 Nonhomogeneous Equations with Constant Coefficients 373
9.5.1 Undetermined Coefficients 373
9.5.2 Variation of Parameters 377
9.5.3 Applications 380
Review Exercises 387
xvl contents
1 0 FOURIER SERIES
10.1 Introduction 390
10.2 Periodicity and Orthogonality of Sines and Cosines 390
10.3 Fourier Series 394
10.4 Convergence of Fourier Series 401
10.5 Fourier Sine and Fourier Cosine Series 409
Review Exercises 418
11 AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS
11.1 Introduction 421
11.2 Definitions and General Comments 423
11.3 The Principle of Superposition 427
11.4 Separation of Variables 433
11.5 Initial-Boundary Value Problems: An Overview 440
11.6 The Homogeneous One-Dimensional Wave Equation: Separation
of Variables 441
11.7 The One-Dimensional Heat Equation 455
11.8 The Potential (Laplace) Equation 461
11.9 Nonhomogeneous Partial Differential Equations: Method I 469
11.10 Nonhomogeneous Partial Differential Equations: Method II 475
Review Exercises 481
APPENDIX A
DETERMINANTS AND LINEAR SYSTEMS OF EQUATIONS AP-1
APPENDIX B
PARTIAL-FRACTION DECOMPOSITION AP-12
APPENDIX C
SOLUTIONS OF POLYNOMIAL EQUATIONS AP-17
APPENDIX D
PROOF OF THE EXISTENCE AND UNIQUENESS THEOREM AP-20
ANSWERS TO ODD-NUMBERED EXERCISES AS-1
INDEX IN-1