Undergraduate Texts in Mathematics Editors S Axler 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 Beardon: Limits: A New Approach to Real Analysis Bak/Newman: Complex Analysis Second edition BanchoffAVermer: Linear Algebra Through Geometry Second edition Berberian: A First Course in Real Analysis Bix: Conies 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 Chambert-Loir: A Field Guide to Algebra 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/Flum/Thomas: Mathematical Logic Second edition Edgar: Measure, Topology, and Fractal Geometry Elaydi: An Introduction to Difference Equations Third edition Erdos/Suranyi: 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 Franklin: Methods of Mathematical Economics (continued after index) J David Logan A First Course in Differential Equations With 55 Figures J David Logan Willa Cather Professor of Mathematics Department of Mathematics University of Nebraska at Lincoln Lincoln, NE 68588-0130 USA dlogan@math.unl.edu Editorial Board S Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA axler@SFSU.edu K.A Ribet Department of Mathematics University of California at Berkeley Berkeley, CA 94720-3840 USA ribet@math.berkeley.edu Mathematics Subject Classification (2000): 34-xx, 15-xx Library of Congress Control Number: 2005926697 (hardcover); Library of Congress Control Number: 2005926698 (softcover) ISBN-10: 0-387-25963-5 (hardcover) ISBN-13: 978-0387-25963-5 ISBN-10: 0-387-25964-3 (softcover) ISBN-13: 978-0387-25964-2 © 2006 Springer Science+Business Media, Inc All 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, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use in this publication of trade 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 Printed in the United States of America springeronline.com (SBA) Dedicated to— Reece Charles Logan, Jaren Logan Golightly Contents Preface xi To the Student xiii Differential Equations and Models 1.1 Differential Equations 1.1.1 Equations and Solutions 1.1.2 Geometrical Interpretation 1.2 Pure Time Equations 1.3 Mathematical Models 1.3.1 Particle Dynamics 1.3.2 Autonomous Differential Equations 1.3.3 Stability and Bifurcation 1.3.4 Heat Transfer 1.3.5 Chemical Reactors 1.3.6 Electric Circuits 2 13 19 21 28 41 45 48 51 Analytic Solutions and Approximations 2.1 Separation of Variables 2.2 First-Order Linear Equations 2.3 Approximation 2.3.1 Picard Iteration 2.3.2 Numerical Methods 2.3.3 Error Analysis 55 55 61 70 71 74 78 viii Contents Second-Order Differential Equations 83 3.1 Particle Mechanics 84 3.2 Linear Equations with Constant Coefficients 87 3.3 The Nonhomogeneous Equation 95 3.3.1 Undetermined Coefficients 96 3.3.2 Resonance 102 3.4 Variable Coefficients 105 3.4.1 Cauchy–Euler Equation 106 3.4.2 Power Series Solutions 109 3.4.3 Reduction of Order 111 3.4.4 Variation of Parameters 112 3.5 Boundary Value Problems and Heat Flow 117 3.6 Higher-Order Equations 124 3.7 Summary and Review 127 Laplace Transforms 133 4.1 Definition and Basic Properties 133 4.2 Initial Value Problems 140 4.3 The Convolution Property 145 4.4 Discontinuous Sources 149 4.5 Point Sources 152 4.6 Table of Laplace Transforms 157 Linear Systems 159 5.1 Introduction 159 5.2 Matrices 165 5.3 Two-Dimensional Systems 179 5.3.1 Solutions and Linear Orbits 179 5.3.2 The Eigenvalue Problem 185 5.3.3 Real Unequal Eigenvalues 187 5.3.4 Complex Eigenvalues 189 5.3.5 Real, Repeated Eigenvalues 191 5.3.6 Stability 194 5.4 Nonhomogeneous Systems 198 5.5 Three-Dimensional Systems 204 Nonlinear Systems 209 6.1 Nonlinear Models 209 6.1.1 Phase Plane Phenomena 209 6.1.2 The Lotka–Volterra Model 217 6.1.3 Holling Functional Responses 221 6.1.4 An Epidemic Model 223 Contents ix 6.2 Numerical Methods 229 6.3 Linearization and Stability 233 6.4 Periodic Solutions 246 6.4.1 The Poincar´e–Bendixson Theorem 249 Appendix A References 255 Appendix B Computer Algebra Systems 257 B.1 Maple 258 B.2 MATLAB 260 Appendix C Sample Examinations 265 Appendix D Solutions and Hints to Selected Exercises 271 Index 287 Preface There are many excellent texts on elementary differential equations designed for the standard sophomore course However, in spite of the fact that most courses are one semester in length, the texts have evolved into calculus-like presentations that include a large collection of methods and applications, packaged with student manuals, and Web-based notes, projects, and supplements All of this comes in several hundred pages of text with busy formats Most students not have the time or desire to read voluminous texts and explore internet supplements The format of this differential equations book is different; it is a one-semester, brief treatment of the basic ideas, models, and solution methods Its limited coverage places it somewhere between an outline and a detailed textbook I have tried to write concisely, to the point, and in plain language Many worked examples and exercises are included A student who works through this primer will have the tools to go to the next level in applying differential equations to problems in engineering, science, and applied mathematics It can give some instructors, who want more concise coverage, an alternative to existing texts The numerical solution of differential equations is a central activity in science and engineering, and it is absolutely necessary to teach students some aspects of scientific computation as early as possible I tried to build in flexibility regarding a computer environment The text allows students to use a calculator or a computer algebra system to solve some problems numerically and symbolically, and templates of MATLAB and Maple programs and commands are given in an appendix The instructor can include as much of this, or as little of this, as he or she desires For many years I have taught this material to students who have had a standard three-semester calculus sequence It was well received by those who xii Preface appreciated having a small, definitive parcel of material to learn Moreover, this text gives students the opportunity to start reading mathematics at a slightly higher level than experienced in pre-calculus and calculus Therefore the book can be a bridge in their progress to study more advanced material at the junior–senior level, where books leave a lot to the reader and are not packaged in elementary formats Chapters 1, 2, 3, 5, and should be covered in order They provide a route to geometric understanding, the phase plane, and the qualitative ideas that are important in differential equations Included are the usual treatments of separable and linear first-order equations, along with second-order linear homogeneous and nonhomogeneous equations There are many applications to ecology, physics, engineering, and other areas These topics will give students key skills in the subject Chapter 4, on Laplace transforms, can be covered at any time after Chapter 3, or even omitted Always an issue in teaching differential equations is how much linear algebra to cover In two extended sections in Chapter we introduce a moderate amount of matrix theory, including solving linear systems, determinants, and the eigenvalue problem In spite of the book’s brevity, it still contains slightly more material than can be comfortably covered in a single three-hour semester course Generally, I assign most of the exercises; hints and solutions for selected problems are given in Appendix D I welcome suggestions, comments, and corrections Contact information is on my Web site: http://www.math.unl.edu/˜dlogan, where additional items may be found I would like to thank John Polking at Rice University for permitting me to use his MATLAB program pplane7 to draw some of the phase plane diagrams and Mark Spencer at Springer for his enthusiastic support of this project Finally, I would like to thank Tess for her continual encouragement and support for my work David Logan Lincoln, Nebraska ...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... Readings in Mathematics Brickman: Mathematical Introduction to Linear Programming and Game Theory Browder: Mathematical Analysis: An Introduction Buchmann: Introduction to Cryptography Buskes/van Rooij:... the qualitative ideas that are important in differential equations Included are the usual treatments of separable and linear first-order equations, along with second-order linear homogeneous and