The Theory of Differential Equations Classical and Qualitative
For over 300 years, differential equations have served as an essential tool for describing and analyzing problems in many scientific disciplines. This carefully-written textbook provides an introduction to many of the important topics associated with ordi
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Walter G. Kelley • Allan C. Peterson
The Theory of Differential Equations Classical and Qualitative Second Edition
Walter G. Kelley Department of Mathematics University of Oklahoma Norman, OK 73019 USA [email protected]
Allan C. Peterson Department of Mathematics University of Nebraska-Lincoln Lincoln, NE 68588-0130 USA [email protected]
Editorial Board: Sheldon Axler, San Francisco State University Vincenzo Capasso, Università degli Studi di Milano Carles Casacuberta, Universitat de Barcelona Angus MacIntyre, Queen Mary, University of London Kenneth Ribet, University of California, Berkeley Claude Sabbah, CNRS, École Polytechnique Endre Süli, University of Oxford ´ Wojbor Woyczynski, Case Western Reserve University
ISBN 978-1-4419-5782-5 e-ISBN 978-1-4419-5783-2 DOI 10.1007/978-1-4419-5783-2 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010924820 Mathematics Subject Classification (2010): 34-XX, 34-01 © Springer Science+Business Media, LLC 2010 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, LLC, 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 on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
We dedicate this book to our families: Marilyn and Joyce and Tina, Carla, David, and Carrie.
Contents Preface
ix
Chapter 1 First-Order Differential Equations 1.1 Basic Results 1.2 First-Order Linear Equations 1.3 Autonomous Equations 1.4 Generalized Logistic Equation 1.5 Bifurcation 1.6 Exercises
1 1 4 5 10 14 16
Chapter 2 Linear Systems 2.1 Introduction 2.2 The Vector Equation x′ = A(t)x 2.3 The Matrix Exponential Function 2.4 Induced Matrix Norm 2.5 Floquet Theory 2.6 Exercises
23 23 27 42 59 64 76 87 87 90 96 107 113
Chapter 3 Autonomous Systems 3.1 Introduction 3.2 Phase Plane Diagrams 3.3 Phase Plane Diagrams for Linear Systems 3.4 Stability of Nonlinear Systems 3.5 Linearization of Nonlinear Systems 3.6 Existence and Nonexistence of Periodic Solutions 3.7 Three-Dimensional Systems 3.8 Differential Equations and Mathematica 3.9 Exercises
120 134 145 149
Chapter 4 Perturbation Methods 4.1 Introduction 4.2 Periodic Solutions 4.3 Singular Perturbations 4.4 Exercises
161 161 172 178 186
Chapter 5
The Self-Adjoint Second-Order Differential Equation 5.1 Basic Definitions
vii
192 192
viii
Contents
5.2 An Interesting Example 5.3 Cauch
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