Table Of Content11
Texts in Applied Mathematics
Editors
JE. Marsden
L. Sirovich
M. Golubitsky
W. Jäger
F. John (deceased)
Advisors
D. Barkley
M. Dellnitz
P. Holmes
G. Iooss
P. Newton
Texts in Applied Mathematics
1. Sirovich: Introduction to Applied Mathematics.
2. Wiggins: Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd ed.
3. Hale/Kor;ak: Dynamics and Bifurcations.
4. Chorin/Marsden: A Mathematical Introduction to Fluid Mechanics, 3rd ed.
5. Hubbard/West: Differential Equations: A Dynamical Systems Approach: Ordinary
Differential Equations.
6. Sontag: Mathematical Control Theory: Deterministic Finite Dimensional Systems,
2nd ed.
7. Perko: Differential Equations and Dynamical Systems, 3rd ed.
8. Seaborn: Hypergeometrie Functions and Their Applications.
9. Pipkin: A Course on Integral Equations.
I 0. Hoppensteadt/Peskin: Modeling and Simulation in Medicine and the Life Sciences,
2nd ed.
II. Braun: Differential Equations and Their Applications, 4th ed.
12. Stoer/Bulirsch: Introduction to Numerical Analysis, 3rd ed.
13. Renardy/Rogers: An Introduction to Partial Differential Equations, 2nd ed.
14. Banks: Growth and Diffusion Phenomena: Mathematical Framewerksand
Applications.
15. Brenner/Scott: The Mathematical Theory ofFinite Element Methods, 2nd ed.
16. Va n de Velde: Concurrent Scientific Computing.
17. Marsden/Ratiu: Introduction to Mechanics and Syrnrnetry, 2nd ed.
18. Hubbard/West: Differential Equations: A Dynamical Systems Approach: Higher-
Dimensional Systems.
19. Kaplan/Glass: Understanding Nonlinear Dynamics.
20. Holmes: Introduction to Perturbation Methods.
21. Curtain/Zwart: An Introduction to Infinite-Dimensional Linear Systems Theory.
22. Thomas: Numerical Partial Differential Equations: Finite Difference Methods.
23. Taylor: Partial Differential Equations: Basic Theory.
24. Merkin: Introduction to the Theory of Stability of Motion.
25. Naher: Topology, Geometry, and Gauge Fields: Foundations.
26. Polderman/Willems: Introduction to Mathematical Systems Theory: A Behavioral
Approach.
27. Reddy: Introductory Functional Analysis with Applications to Boundary-Value
Problems and Finite Elements.
28. Gustafson/Wilcox: Analytical and Computational Methods of Advanced
Engineering Mathematics.
29. Tveito/Winther: Introduction to Partial Differential Equations: A Computational
Approach.
30. Gasquet/Witomski: Fourier Analysis and Applications: Filtering, Numerical
Computation, Wavelets.
(continued after index)
Martin Braun
Differential Equations
and Their Applications
An Introduction to
Applied Mathematics
F ourth Edition
With 68 Illustrations
~Springer
Martin Braun
Department of Mathematics
Queens College
City University of New York
Flushing, NY 11367
USA
Series Editors
Jerrold E. Marsden L. Sirovich
Control and Dynamical Systems, 107-81 Division of App!ied Mathematics
California Institute of Technology Brown University
Pasadena, CA 91125 Providence, RI 02912
USA USA
M. Golubitsky W. Jăger
Department of Mathematics Department of Applied Mathematics
University of Houston Universităt Heidelberg
Houston, TX 77204-3476 Im Neuenheimer Feld 294
USA 69120 Heidelberg, Germany
Mathematics Subject Classification (1991): 34-01
Library of Congress Cata1oging-in-Pub1ication Data
Braun, Martin, 1941-
Differentia1 equations and their app1ications: an introduction to
applied mathematics / M. Braun.-4th ed.
p. cm.-(Texts in app1ied mathematics; 11)
Includes bibliographical references and index.
ISBN 978-0-387-94330-5 ISBN 978-1-4612-4360-1 (eBook)
DOI 10.1007/978-1-4612-4360-1
1. Differential equations. 1. Title. Il. Series.
QA37l.B795 1992
515'.35-dc20 92-24317
ISBN 978-0-387-94330-5 Printed on acid-free paper.
© 1993 Springer Science+Business Media New York
Softcover reprint of the hardcover 1s t edition 1993.
Ali rights reserved. This work may not be translated or copied in whole or in part without the
written permission of the publisher (Springer Science+Business Media, LLC)
except for brief excerpts in connection with reviews or scholarly analysis.
Use in connection with any form of information storage and retrieval, electronic
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To Jour beautiful people:
Zelda Lee
Adeena Rachelle, I. Nasanayl, and Shulamit
Series Preface
Mathematics is playing an ever more important role in the physical and
biological sciences, provoking a blurring of boundaries between scientific
disciplines and a resurgence of interest in the modern as weil as the classical
techniques of applied mathematics. This renewal of interest, both in research
and teaching, has led to the establishment of the series: Texts in Applied
Mathematics (TAM).
The development of new courses is a natural consequence of a high Ievel
of excitement on the research frontier as newer techniques, such as numerical
and symbolic computer systems, dynamical systems, and chaos, mix with and
reinforce the traditional methods of applied mathematics. Thus, the purpose
of this textbook series is to meet the current and future needs of these
advances and encourage the teaching of new courses.
T AM will publish textbooks suitable for use in advanced undergraduate
and beginning graduate courses, and will complement the Applied Mathe
matical Seiences ( AMS) series, which will focus on advanced textbooks and
research Ievel monographs.
Preface to the Fourth Edition
There are two major changes in the Fourth Edition of Differential Equations
and Their Applications. The first concerns the computer programs in this text.
In keeping with recent trends in computer science, we have replaced all the
APL programs with Pascal and C programs. The Pascal programs appear in
the text in place ofthe APL programs, where they are followed by the Fortran
programs, while the C programs appear in Appendix C.
The second change, in response to many readers' suggestions, is the in
clusion of a new chapter (Chapter 6) on Sturm-Liouville boundary value
problems. Our goal in this chapter is not to present a whole lot of technical
material. Rather it is to show that the theory of Fourier series presented in
Chapter 5 is not an isolated theory but is part of a much more general and
beautiful theory which encompasses many of the key ideas of linear algebra.
To accomplish this goal we have included some additional material from
linear algebra. In particular, we have introduced the notions of inner product
spaces and self-adjoint matrices, proven that the eigenvalues of a self-adjoint
matrix are real, and shown that all self-adjoint matrices possess an ortho
normal basis of eigenvectors. These results are at the heart of Sturm-Liouville
theory.
I wish to thank Robert Giresi for writing the Pascal and C programs.
New York City Martin Braun
May, 1992
Preface to the Third Edition
There are three major changes in the Third Edition of Differential Equations
and Their Applications. First, we have completely rewritten the section on
singular solutions of differential equations. A new section, 2.8.1, dealing
with Euler equations has been added, and this section is used to motivate a
greatly expanded treatment of singular equations in sections 2.8.2 and 2.8.3.
Our second major change is the addition of a new section, 4.9, dealing
with bifurcation theory, a subject of much current interest. We felt it
desirable to give the reader a brief but nontrivial introduction to this
important topic.
Our third major change is in Section 2.6, where we have switched to the
metric system of units. This change was requested by many of our readers.
In addition to the above changes, we have updated the material on
population models, and have revised the exercises in this section. Minor
editorial changes have also been made throughout the text.
New York City
November, 1982 Martin Braun
Preface to the First Edition
This textbook is a unique blend of the theory of differential equations· and
their exciting application to "real world" problems. First, and foremost, it
is a rigorous study of ordinary differential equations and can be fully
understood by anyone who has completed one year of calculus. However,
in addition to the traditional applications, it also contains many exciting
"real life" problems. These applications are completely self contained.
First, the problern to be solved is outlined clearly, and one or more
differential equations are derived as a model for this problem. These
equations are then solved, and the results are compared with real world
data. The following applications are covered in this text.
1. In Section 1.3 we prove that the beautiful painting "Disciples of
Emmaus" which was bought by the Rembrandt Society of Belgium for
$170,000 was a modern forgery.
2. In Section 1.5 we derive differential equations which govem the
population growth of various species, and compare the results predicted by
our models with the known values of the populations.
3. In Section 1.6 we derive differential equations which govern the rate at
which farmers adopt new innovations. Surprisingly, these same differential
equations govem the rate at which technological innovations are adopted in
such diverse industries as coal, iron and steel, brewing, and railroads.
4. In Section 1.7 we try to determine whether tightly sealed drums filled
with concentrated waste material will crack upon impact with the ocean
floor. In this section we also describe several tricks for obtaining informa
tion about solutions of a differential equation that cannot be solved
explicitly.
