Table Of Content

Undergraduate Texts in Mathematics Editors S. Axler F.W. Gehring K.A. Ribet Undergraduate Texts in Mathematics Abbott: Understanding Analysis. Childs: A Concrete Introduction to Anglin: Mathematics: A Concise History Higher Algebra. Second edition. and Philosophy. Chung/AitSahlia: Elementary Probability Readings in Mathematics. Theory: With Stochastic Processes and Anglin/Lambek: The Heritage of an Introduction to Mathematical Thales. Finance. Fourth edition. Readings in Mathematics. Cox/Little/O'Shea: Ideals, Varieties, Apostol: Introduction to Analytic and Algorithms. Second edition. Number Theory. Second edition. Croom: Basic Concepts of Algebraic Armstrong: Basic Topology. Topology. Armstrong: Groups and Symmetry. Curtis: Linear Algebra: An Introductory Axler: Linear Algebra Done Right. Approach. Fourth edition. Second edition. Daepp/Gorkin: Reading, Writing, and Reardon: Limits: A New Approach to Proving: A Closer Look at Real Analysis. Mathematics. Bak!Newman: Complex Analysis. Devlin: The Joy of Sets: Fundamentals Second edition. of Contemporary Set Theory. Banchoff/Wermer: Linear Algebra Second edition. Through Geometry. Second edition. Dixmier: General Topology. Berberian: A First Course in Real Driver: Why Math? Analysis. Ebbinghaus/Fium/Thomas: Bix: Conics and Cubics: A Mathematical Logic. Second edition. Concrete Introduction to Algebraic Edgar: Measure, Topology, and Fractal Curves. Geometry. Bremaud: An Introduction to Elaydi: An Introduction to Difference Probabilistic Modeling. Equations. Second edition. Bressoud: Factorization and Primality Erdi:ls/Suninyi: Topics in the Theory of Testing. Numbers. Bressoud: Second Year Calculus. Estep: Practical Analysis in One Variable. Readings in Mathematics. Exner: An Accompaniment to Higher Brickman: Mathematical Introduction Mathematics. to Linear Programming and Game Exner: Inside Calculus. Theory. Fine/Rosenberger: The Fundamental Browder: Mathematical Analysis: Theory of Algebra. An Introduction. Fischer: Intermediate Real Analysis. Buchmann: Introduction to Flanigan/Kazdan: Calculus Two: Linear Cryptography. and Nonlinear Functions. Second Buskes/van Rooij: Topological Spaces: edition. From Distance to Neighborhood. Fleming: Functions of Several Variables. Callahan: The Geometry of Spacetime: Second edition. An Introduction to Special and General Foulds: Combinatorial Optimization for Relavitity. Undergraduates. Carter/van Brunt: The Lebesgue Foulds: Optimization Techniques: An Stieltjes Integral: A Practical Introduction. Introduction. (continued after index) Cederberg: A Course in Modern Geometries. Second edition. Clay C. Ross Differential Equations An Introduction with Mathematica® Second Edition With 88 Figures ~Springer Clay C. Ross Department of Mathematics The University of the South 735 University Avenue Sewanee, TN 37383 USA [email protected] Editorial Board S. Axler F.W. Gehring K.A. Ribet Mathematics Department Mathematics Department Mathematics Department San Francisco State East Hali University of California, University University of Michigan Berkeley San Francisco, CA 94132 Ann Arbor, MI 48109 Berkeley, CA 947.20-3840 USA USA USA Mathematics Subject Classification (2000): 34-01, 65Lxx Library of Congress Cataloging-in-Publication Data Ross, Clay C. Differential equations : an introduction with Mathematica / Clay C. Ross. - 2nd ed. p. cm. - (Undergraduate texts in mathematics) Includes bibliographical references and index. ISBN 978-1-4419-1941-0 ISBN 978-1-4757-3949-7 (eBook) DOI 10.1007/978-1-4757-3949-7 1. Differential equations. 2. Mathematica (Computer file) 1. Title. TI. Series. QA371.R595 2004 515'.35-dc22 2004048203 ISBN 978-1-4419-1941-0 Printed on acid-free paper. © 2004, 1995 Spinger Science+Business Media New York Originally published by Springer Science+ Business Media , loc. in 2004 Softcover reprint of the hardcover lod edition 2004 AII rights reserved. This work may not be translated or copied in whole or in part with out the written permission of the publisher Springer Science+Business Media, u.e , except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any farm of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar meth odology now known or hereafter developed is farbidden. The use in this publication oftrade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. (MVY) 9 8 7 6 543 2 1 SPIN 10963685 springeronline.com This book is dedicated to my parents, Vera K. and Clay C. Ross, and to my wife, Andrea, with special gratitude to each. Preface Goals and Emphasis of the Book Mathematicians have begun to find productive ways to incorporate computing power into the mathematics curriculum. There is no attempt here to use computing to avoid doing differential equations and linear algebra. The goal is to make some first ex plorations in the subject accessible to students who have had one year of calculus. Some of the sciences are now using the symbol-manipulative power of Mathemat ica to make more of their subject accessible. This book is one way of doing so for differential equations and linear algebra. I believe that if a student's first exposure to a subject is pleasant and exciting, then that student will seek out ways to continue the study of the subject. The theory of differential equations and of linear algebra permeates the discussion. Every topic is supported by a statement of the theory. But the primary thrust here is obtaining solutions and information about solutions, rather than proving theorems. There are other courses where proving theorems is central. The goals of this text are to establish a solid understanding of the notion of solution, and an appreciation for the confidence that the theory gives during a search for solutions. Later the student can have the same confidence while personally developing the theory. When a study of the book has been completed, many important elementary con cepts of differential equations and linear algebra will have been encountered. In addition, the use of Mathematica makes it possible to analyze problems that are formidable without computational assistance. Mathematica is an integral part of the presentation, because in introductory differential equations or linear algebra courses it is too often true that simple tasks like finding an antiderivative, or finding the roots of a polynomial of relatively high degree-even when the roots are all rational completely obscure the mathematics that is being studied. The complications en countered in the manual solution of a realistic problem of four first-order linear equa tions with constant coefficients can totally obscure the beauty and centrality of the theory. But having Mathematica available to carry out the complicated steps frees the student to think about what is happening, how the ideas work together, and what everything means. VIII Preface The text contains many examples. Most are followed immediately by the same example done in Mathematica. The form of a Mathematica notebook is reproduced almost exactly so that the student knows what to expect when trying problems by him/herself. Having solutions by Mathematica included in the text also provides a sort of encyclopedia of working approaches to doing things in Mathernatica. In ad dition, each of these examples exists as a real Mathematica notebook that can be executed, studied, printed out, or modified to do some other problem. Other Math ematica notebooks may be provided by the instructor. Occasionally a problem will request that new methods be tried, but by the time these occur, students should be able to write effective Mathematica code of their own. Mathematica can carry the bulk of the computational burden, but this does not relieve the student of knowing whether or not what is being done is correct. For that reason, periodic checking of results is stressed. Often an independent manual calcu lation will keep a Mathematica calculation safely on course. Mathematica, itself, can and should do much of the checking, because as the problems get more complex, the calculations get more and more complicated. A calculation that is internally consis tent stands a good chance of being correct when the concepts that are guiding the process are correct. Since all of the problems except those that are of a theoretical nature can be solved and checked in Mathematica, very few of the exercises have answers supplied. As the student solves the problems in each section, they should save the notebooks to disk-where they can serve as an answer book and study guide if the solutions have been properly checked. A Mathematica package is a collection of functions that are designed to perform certain operations. Several notebooks depend heavily on a package that has been provided. Most of the packages supplied undertake very complicated tasks, where the functions are genuinely intimidating, so the code does not appear in the text of study notebooks. What Is New in This Edition The changes are two-fold: 1. Rearrange and restate some topics (Linear algebra has now been gathered into a separate chapter, and series methods for systems have been eliminated.) Many typographical errors have been corrected. 2. Completely rewrite, and occasionally expand, the Mathematica code using ver sion 5 of Mathematica. In addition, since Mathematica now includes a complete and fully on-line Help sub system, several appendices have been eliminated. Topics Receiving Lesser Emphasis The solutions of most differential equations are not simple, so the solutions of such equations are often examined numerically. We indicate some ways to have Math ematica solve differential equations numerically. Also, properties of a solution are Preface IX often deduced from careful examination of the differential equation itself, but an extended study of qualitative differential equations must wait for a more advanced course. The best advice is to use the NDSol ve function when a numerical solution is required. Some differential equations have solutions that are very hard to describe either analytically or numerically because the equations are sensitive to small changes in the initial values. Chaotic behavior is a topic of great current interest; we present some examples of such equations, but do not fully develop the concepts. Acknowledgments I would like to thank those students and others who read the manuscript for the first edition, and several science department colleagues for enduring questions and for responding so kindly. Reviews of the first edition were received from Professors Matthew Richey, Margie Hale, Stephen L. Clark, Stan Wagon, and William Sit. Dave Withoff con tributed to that edition his expert help on technical aspects of Mathematica program ming. Any errors that remain in this edition are solely the responsibility of the author. Sewanee, Tennessee, USA CLAY C. ROSS November 2003 Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII 1 About Differential Equations ................................... . 1.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2 Uniqueness Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.3 Differential Inclusions (Optional) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2 Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.1 Familiar Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2 Abstract Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.3 Differential Equations from Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.4 Characteristic Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3 First-Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.1 First-Order Linear Differential Equations. . . . . . . . . . . . . . . . . . . . . . . 52 3.2 Linear Equations by M athematica . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.3 Exact Equations............................................ 59 3.4 Variables Separable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.5 Homogeneous Nonlinear Differential Equations . . . . . . . . . . . . . . . . . 75 3.6 Bernoulli and Riccati Differential Equations (Optional)........... 79 3.7 Clairaut Differential Equations (Optional) . . . . . . . . . . . . . . . . . . . . . . 86 4 Applications of First-Order Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.1 Orthogonal Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.2 Linear Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.3 Nonlinear Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 XII Contents 5 Higher-Order Linear Differential Equations . . . . . . . . . . . . . . . . . . . . . . 129 5.0 Introduction ............................................... 129 5.1 The Fundamental Theorem .................................. 130 5.2 Homogeneous Second-Order Linear Constant Coefficients ........ 139 5.3 Higher-Order Constant Coefficients (Homogeneous) ............. 152 5.4 The Method of Undetermined Coefficients ..................... 160 5.5 Variation of Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 6 Applications of Second-Order Equations. . . . . . . . . . . . . . . . . . . . . . . . . . 179 6.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 6.1 Simple Harmonic Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 6.2 Damped Harmonic Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 6.3 Forced Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 6.4 Simple Electronic Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 6.5 Two Nonlinear Examples (Optional). . . . . . . . . . . . . . . . . . . . . . . . . . . 206 7 The Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 7.0 Introduction ............................................... 210 7.1 The Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 7.2 Properties of the Laplace Transform ........................... 214 7.3 The Inverse Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 7.4 Discontinous Functions and Their Transforms. . . . . . . . . . . . . . . . . . . 230 8 Higher-Order Differential Equations with Variable Coefficients . . . . . 240 8.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 8.1 Cauchy-Euler Differential Equations .......................... 241 8.2 Obtaining a Second Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 8.3 Sums, Products and Recursion Relations . . . . . . . . . . . . . . . . . . . . . . . 253 8.4 Series Solutions of Differential Equations . . . . . . . . . . . . . . . . . . . . . . 262 8.5 Series Solutions About Ordinary Points . . . . . . . . . . . . . . . . . . . . . . . . 269 8.6 Series Solution About Regular Singular Points . . . . . . . . . . . . . . . . . . 277 8.7 Important Classical Differential Equations and Functions ......... 293 9 Differential Systems: Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 9.0 Introduction ............................................... 297 9.1 Reduction to First-Order Systems ............................. 302 9.2 Theory of First-Order Systems ............................... 309 9.3 First- Order Constant Coefficients Systems . . . . . . . . . . . . . . . . . . . . . . 317 9.4 Repeated and Complex Roots ................................ 331 9.5 Nonhomogeneous Equations and Boundary-Value Problems ...... 344 9.6 Cauchy-Euler Systems ...................................... 358