11 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. 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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.