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Undergraduate Texts in Mathematics Editors S Axler F.W Gehring K.A Ribet Undergraduate Texts in Mathematics Abbott: Understanding Analysis Anglin: Mathematics: A Concise History and Philosophy Readings in Mathematics Anglin/Lambek: The Heritage of Thales Readings in Mathematics Apostol: Introduction to Analytic Number Theory Second edition Armstrong: Basic Topology Armstrong: Groups and Symmetry Axler: Linear Algebra Done Right Second edition Reardon: Limits: A New Approach to Real Analysis Bak!Newman: Complex Analysis Second edition Banchoff/Wermer: Linear Algebra Through Geometry Second edition Berberian: A First Course in Real Analysis Bix: Conics and Cubics: A Concrete Introduction to Algebraic Curves Bremaud: An Introduction to Probabilistic Modeling Bressoud: Factorization and Primality Testing Bressoud: Second Year Calculus Readings in Mathematics Brickman: Mathematical Introduction to Linear Programming and Game Theory Browder: Mathematical Analysis: An Introduction Buchmann: Introduction to Cryptography Buskes/van Rooij: Topological Spaces: From Distance to Neighborhood Callahan: The Geometry of Spacetime: An Introduction to Special and General Relavitity Carter/van Brunt: The LebesgueStieltjes Integral: A Practical Introduction Cederberg: A Course in Modern Geometries Second edition Childs: A Concrete Introduction to Higher Algebra Second edition Chung/AitSahlia: Elementary Probability Theory: With Stochastic Processes and an Introduction to Mathematical Finance Fourth edition Cox/Little/O'Shea: Ideals, Varieties, and Algorithms Second edition Croom: Basic Concepts of Algebraic Topology Curtis: Linear Algebra: An Introductory Approach Fourth edition Daepp/Gorkin: Reading, Writing, and Proving: A Closer Look at Mathematics Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory Second edition Dixmier: General Topology Driver: Why Math? Ebbinghaus/Fium/Thomas: Mathematical Logic Second edition Edgar: Measure, Topology, and Fractal Geometry Elaydi: An Introduction to Difference Equations Second edition Erdi:ls/Suninyi: Topics in the Theory of Numbers Estep: Practical Analysis in One Variable Exner: An Accompaniment to Higher Mathematics Exner: Inside Calculus Fine/Rosenberger: The Fundamental Theory of Algebra Fischer: Intermediate Real Analysis Flanigan/Kazdan: Calculus Two: Linear and Nonlinear Functions Second edition Fleming: Functions of Several Variables Second edition Foulds: Combinatorial Optimization for Undergraduates Foulds: Optimization Techniques: An Introduction (continued after index) 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 cross@Jsewanee.edu Editorial Board S Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA F.W Gehring Mathematics Department East Hali University of Michigan Ann Arbor, MI 48109 USA K.A Ribet Mathematics Department University of California, Berkeley Berkeley, CA 947.20-3840 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 Differential equations QA371.R595 2004 515'.35-dc22 ISBN 978-1-4419-1941-0 Mathematica (Computer file) Title TI Series 2004048203 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 without 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 methodology 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 expressio n of opinion as to whether or not they are subject to proprietary rights (MVY) 543 springeronline.com SPIN 10963685 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 explorations 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 Mathematica 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 concepts 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 rationalcompletely obscure the mathematics that is being studied The complications encountered in the manual solution of a realistic problem of four first-order linear equations 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 addition, each of these examples exists as a real Mathematica notebook that can be executed, studied, printed out, or modified to some other problem Other Mathematica 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 calculation will keep a Mathematica calculation safely on course Mathematica, itself, can and should much of the checking, because as the problems get more complex, the calculations get more and more complicated A calculation that is internally consistent 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: 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 Completely rewrite, and occasionally expand, the Mathematica code using version of Mathematica In addition, since Mathematica now includes a complete and fully on-line Help subsystem, 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 Mathematica 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 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 contributed to that edition his expert help on technical aspects of Mathematica programming Any errors that remain in this edition are solely the responsibility of the author Sewanee, Tennessee, USA November 2003 CLAY C ROSS Contents Preface VII About Differential Equations 1.0 Introduction 1.1 Numerical Methods 1.2 Uniqueness Considerations 1.3 Differential Inclusions (Optional) 17 22 Linear Algebra 2.0 Introduction 2.1 Familiar Linear Spaces 2.2 Abstract Linear Spaces 2.3 Differential Equations from Solutions 2.4 Characteristic Value Problems 26 26 31 31 44 48 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 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 Higher-Order Linear Differential Equations 5.0 Introduction 5.1 The Fundamental Theorem 5.2 Homogeneous Second-Order Linear Constant Coefficients 5.3 Higher-Order Constant Coefficients (Homogeneous) 5.4 The Method of Undetermined Coefficients 5.5 Variation of Parameters 129 129 130 139 152 160 171 Applications of Second-Order Equations 6.0 Introduction 6.1 Simple Harmonic Motion 6.2 Damped Harmonic Motion 6.3 Forced Oscillation 6.4 Simple Electronic Circuits 6.5 Two Nonlinear Examples (Optional) 179 179 179 190 197 202 206 The Laplace Transform 7.0 Introduction 7.1 The Laplace Transform 7.2 Properties of the Laplace Transform 7.3 The Inverse Laplace Transform 7.4 Discontinous Functions and Their Transforms 210 210 211 214 225 230 Higher-Order Differential Equations with Variable Coefficients 8.0 Introduction 8.1 Cauchy-Euler Differential Equations 8.2 Obtaining a Second Solution 8.3 Sums, Products and Recursion Relations 8.4 Series Solutions of Differential Equations 8.5 Series Solutions About Ordinary Points 8.6 Series Solution About Regular Singular Points 8.7 Important Classical Differential Equations and Functions 240 240 241 251 253 262 269 277 293 Differential Systems: Theory 9.0 Introduction 9.1 Reduction to First-Order Systems 9.2 Theory of First-Order Systems 9.3 First-Order Constant Coefficients Systems 9.4 Repeated and Complex Roots 9.5 Nonhomogeneous Equations and Boundary-Value Problems 9.6 Cauchy-Euler Systems 297 297 302 309 317 331 344 358 10.5 Solution of Linear Systems by Laplace Transforms (Optional) 419 Example 10.7 (M) Use the function LP T So ve provided in the package LPT.m to solve the initial value problem stated in Example 10.7 Solution Assume the LP T m package is loaded In[6] := Clear[y, z] In[7]:= LPTSolve[{y"[t] -z[t] ==0, z'[t] +y[t] == 0, y[O] == 1, y' [0] == 0, z[0]==0}, {y[t], z [t] }, t, s] The transformed system { -5 + LaplaceTransform [y [ t], t, 5]- LaplaceTransform [ z [ t], t, 5] == 0, LaplaceTransform [y [ t], t, 5] + LaplaceTransform [ z [ t], t, 5] == 0} The unknown ( s) isolated { LaplaceTransform [y [ t], t, 52 5] -7 - , 1+5 LaplaceTransform [ z [ t] , t, 5] Out [7]= {y [t] -7 ~ z [ t] -) - ( e-t + ~ e-t et 12 ( - -7 - ~} v; l), v; 1+5 Cos [ + e t/2 t (Cos [ t ] + V3 Sin [ ~ t ]) ) } In[B] := {y[t_], z[t_]} = {y[t], z[t] }/.% Out[B]= {~ (e-t+2et 12 cos[~tJ), _ ~ e-t ( _ + e3 t/2 (Cos [ v; t] + V3 Sin [ v; t] ) ) } In[9]:= Simplify[{y"[t] -z[t] ==0, z'[t] +y[t] == 0, y[O] == 1, y' [0] == 0, z[O] ==0}] Out[9]= {True, True, True, True, True} Example 10.8 (M) Define f(t) = { 1, 0, ~t~2 2 e-· ( - + e- +¥ (Cos [ ~ ~ ( -2 + t)] + ~ Sin [ ~ ~ ( -2 + t)])) Unit Step [ -2 + t]} In[l3] := {y[t_], z[t_)} = {y[t], z[t] }/.% Out[13]= {1 + ~ (- + e -t + 2e- +~ Cos [ ~~ (-2 + t)] )unitStep[ -2 + t], e 2-t -3 (- + e- + (Cos l~ ~ (-2 + t)] + ~ Sin [ ~ VJ (-2 + t)])) UnitStep[-2+tJ} 10.5 Solution of Linear Systems by Laplace Transforms (Optional) 421 In[14]:= Simplify[{y"(t] -z[t] ==0, z'[t] +y[t] == f[t], y[O] == 1, y' (0] == 0, z[O] == 0}] Out[14]= {True, True, True, True, True} A close look at this solution shows that it has a two-part definition with the break at t = If you run this example for yourself, you will find that it takes a while to run to completion But imagine doing it by hand! This brief introduction to the use of Laplace transforms to solve nonhomogeneous initial value problems should help convince you that the Laplace transform is a powerful tool As it has been introduced, the Laplace transform is applicable only to initial value problems for ordinary differential equations and systems, but they are very common problems The Laplace transform has other applications, as well Exercises 10.4 Use the function LPTSol ve from the package LPT.m to solve these initial value problems for systems -y 1(t) + 2y (t) + 2y (t), 2y 1(t) + 2y (t) + 2y (t), -3y 1(t)- 6y 2(t)- 6y 3(t); 1, y2 (0) = -3, y (0) = 2y (t) 2y, (t) 1, y2 (0) = -y 1(t) + 2y2 (t) + 2y3 (t), 2y 1(t) + 2y2 (t) + 2y3 (t) + 1, -3y 1(t)- 6y2 (t)- 6y (t); 1, y (0) = -3, y (0) = 2yz(t), 2y, (t); 0, y (0) = -y,(t) + 2yz(t) + 2y3(t), 2y 1(t) + 2y2 (t) + 2y3 (t) + 1, - 3y 1(t) - 6y (t)- 6y (t) +sin 5t; 1, y (0) = 0, y (0) = 2yz(t) + e3r, 2y 1(t)-e ; 1, y (0) = L et !( t ) = ( e , ' ~2 t ~

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