UNITEXT La Matematica per il 3+2 Volume 73 For further volumes: http://www.springer.com/series/5418
Shair Ahmad Antonio Ambrosetti A Textbook on Ordinary Differential Equations
Shair Ahmad Department of Mathematics University of Texas at San Antonio San Antonio, USA Antonio Ambrosetti SISSA Trieste, Italy UNITEXT La Matematica per il 3+2 ISSN 2038-5722 ISSN 2038-5757 (electronic) ISBN 978-3-319-02128-7 ISBN 978-3-319-02129-4 (ebook) DOI 10.1007/978-3-319-02129-4 Springer Cham Heidelberg New York Dordrecht London Library of Congress Control Number: 2013945784 Springer International Publishing Switzerland 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Cover Design: Beatrice B, Milano Typesetting with LATEX: PTP-Berlin, Protago TEX-Production GmbH, Germany (www.ptp-berlin.de) Printing and Binding: GECA Industrie Grafiche, San Giuliano Milanese (MI), Italy Springer is a part of Springer Science+Business Media (www.springer.com)
The first author wishes to thank his wife Carol for her continued and loving support, patience and understanding far beyond what might be normally expected
Preface One of the authors main motivation for writing this book has been to provide students and faculty with a more economical option for selecting a textbook on introduction to ODE. This book is a primer for the theory and applications of Ordinary Differential Equations. It is aimed at students of Mathematics, Physics, Engineering, Statistics, Information Science, etc. with sufficient knowledge of Calculus and a minimal knowledge of Linear Algebra. The first chapter starts with the simplest first order linear differential equations and builds on it to lead to the more general equations. The concepts of initial values and existence and uniqueness of solutions are introduced early in this chapter. Ample examples, using simple integration, are given to motivate and demonstrate these concepts. Almost all of the assertions are proved in elementary and simple terms. The important concepts of the Cauchy Problem and Existence and Uniqueness of solutions are introduced in detail and demonstrated by many examples. Proofs are given in an Appendix. There is also a rigorous treatment of some qualitative behavior of solutions. This chapter is important from a pedagogical point of view because it introduces students to rigor and understanding of important concepts at an early stage. There is also a chapter on nonlinear first order equations, where students learn how to explicitly solve certain types of equations such as separable, homogeneous, exact, Bernoulli and Clairaut equations. Further chapters are devoted to linear higher order equations and systems, with several applications to mechanics and electrical circuit theory. Also included is an elementary but rigorous introduction to the theory of oscillation. There is a chapter on phase plane analysis dealing with finding periodic solutions, solutions of simple boundary value problems, homoclinic and heteroclinic trajectories. There is also a section discussing a Lotka Volterra system arising in population dynamics. Subsequently, the book deals with the Sturm Liouville eigenvalues, Laplace transform and finding series solutions, including fairly detailed treatment of Bessel functions, which are important in Engineering. Although this book is mainly addressed at undergraduate students, there are some more advanced topics such as stability theory and existence of solutions to Boundary
viii Preface Value problems, which might be useful for the more motivated undergraduates or even beginning graduate students. A chapter on numerical methods is included as an Appendix, where the importance of computer technology is pointed out. Otherwise, we do not encourage the use of computer technology at this level. Besides, we believe that, at this stage, students should practice their previous knowledge of Algebra and Calculus instead of relying on technology; thus sharpening their mathematical skills in general. Each chapter ends with a set of exercises, which are meant to test the students understanding of the concepts covered. Solutions to selected exercises are included at the end of the book. We wish to acknowledge with gratitude the help of Dung Le, Rahbar Maghsoudi, and Vittorio Coti Zelati, especially with technical issues. San Antonio and Trieste December 2013 Shair Ahmad Antonio Ambrosetti
Notation The following are some notations that are used in the book. N denotes the set of natural numbers 0; 1; 2 : : : Z denotes the set of integer numbers 0; 1; 2::: R denotes the set of real numbers. C denotes the set of complex numbers. If a; b 2 R, Œa; b denotes the closed interval ¹a t bº;.a; b/, or a; bœ, denotes the open interval ¹a <t<bº. Moreover.a; b, or a; b, denotes the interval ¹a <t bº, while Œa; b/,orœa; bœ, denotes the interval ¹a t<bº. If x;y 2 R n,.x j y/ D P x i y i denotes the euclidean scalar product of the vectors x;y, with components x i ;y i, i D 1;:::;n. In some case we will also use x y or.x;y/ instead of.x j y/. The corresponding euclidean norm is denoted by jxj D p q P.x j x/ D x 2 i.ifn D 1 then jxj is the usual absolute value. d k f D f.k/ denotes the k-th derivative of f.t/. dt k @f @x i D @ xi f D f xi denotes the partial derivative of f.x 1 ;:::;x n / with respect to x i. If R n, C.;R/,orsimplyC./, is the class of continuous real valued functions f W 7! R defined on. C.;R m / is the class of continuous functions f defined on with values in R m. If R n is an open set, C k.; R/,orsimplyC k./, is the class of real valued functions f W 7! R which are k times continuously differentiable. C.;R m / is the class of functions f W 7! R m, each component of which is k times continuously differentiable. Functions that are differentiable infinitely many times are often called regular. W.f 1 ;:::;f n /.t/ D W.f 1.t/; : : : ; f n.t// D W.t/ represents the Wronskian of the functions f 1 ;:::;f n. J m = Bessel function of order m.
x Notation f g = convolution of the functions f and g. ı.t/ = the Dirac delta function. Det(A) = determinant of the matrix A. A kl = Minor of the element a kl, C kl = cofactor of the element a kl. L¹f.t/º.s/ D F.s/= the Laplace transform of the function f. rv.x/ D.V x1.x/;:::;v xn.x//, x 2 R n, denotes the gradient of the real valued function V..rV.x/ j f.x// D P n 1 V x i.x/f xi.x/ = scalar product of rv.x/ and f.x/.
Contents 1 First order linear differential equations... 1 1.1 Introduction.... 1 1.2 Asimplecase... 2 1.3 Someexamplesarisinginapplications... 3 1.3.1 Population dynamics...... 3 1.3.2 An RC electric circuit..... 4 1.4 Thegeneralcase... 5 1.5 Exercises... 13 2 Theory of first order differential equations... 15 2.1 Differentialequationsandtheirsolutions... 15 2.2 TheCauchyproblem:Existenceanduniqueness... 18 2.2.1 Local existence and uniqueness... 18 2.2.2 Global existence and uniqueness... 24 2.3 Qualitative properties of solutions...... 25 2.4 Improvingtheexistenceanduniquenessresults... 27 2.5 Appendix:Proofofexistenceanduniquenesstheorems... 29 2.5.1 Proof of Theorem 2.4.5.... 29 2.5.2 Proof of Theorem 2.4.4.... 32 2.6 Exercises... 33 3 First order nonlinear differential equations... 35 3.1 Separableequations... 35 3.1.1 The logistic equation...... 37 3.2 Exactequations... 39 3.3 Theintegratingfactor... 49 3.4 Homogeneousequations... 52 3.5 Bernoulliequations... 56 3.6 Appendix. Singular solutions and Clairaut equations... 57 3.6.1 Clairaut equations.... 59 3.7 Exercises... 62
xii Contents 4 Existence and uniqueness for systems and higher order equations... 65 4.1 Systemsofdifferentialequations... 65 4.1.1 Existence and uniqueness results for systems... 67 4.2 Higherorderequations... 68 4.2.1 Existence and uniqueness for n-thorderequations... 69 4.3 Exercises... 70 5 Second order equations... 71 5.1 Linearhomogeneousequations... 71 5.2 LinearindependenceandtheWronskian... 75 5.2.1 Wronskian.... 77 5.3 Reductionoftheorder... 80 5.4 Linear nonhomogeneous equations...... 81 5.4.1 Variation of parameters.... 83 5.5 Linearhomogeneousequationswithconstantcoefficients... 85 5.5.1 The Euler equation.... 92 5.6 Linear nonhomogeneous equations method of undetermined coefficients... 93 5.6.1 The elastic spring... 97 5.7 Oscillatory behavior of solutions.... 99 5.8 Somenonlinearsecondorderequations... 105 5.8.1 Equations of the type F.t;x 0 ;x 00 / D 0... 105 5.8.2 Equations of the type F.x;x 0 ;x 00 / D 0... 105 5.8.3 Equations of the form F.t;x;x 0 ;x 00 / D 0 with F homogenous106 5.9 Exercises... 108 6 Higher order linear equations... 113 6.1 Existenceanduniqueness... 113 6.2 LinearindependenceandWronskian... 114 6.3 Constantcoefficients... 115 6.4 Nonhomogeneous equations...... 118 6.5 Exercises... 121 7 Systems of first order equations... 123 7.1 Preliminaries:Abriefreviewoflinearalgebra... 123 7.1.1 Basic properties of matrices..... 123 7.1.2 Determinants....... 124 7.1.3 Inverse of a matrix... 127 7.1.4 Eigenvalues and eigenvectors.... 128 7.1.5 The Jordan normal form... 130 7.2 Firstordersystems... 132 7.3 Linearfirstordersystems... 134 7.3.1 Wronskian and linear independence..... 136 7.4 Constantsystems eigenvaluesandeigenvectors... 140 7.5 Nonhomogeneous systems... 146 7.6 Exercises... 150
Contents xiii 8 Qualitative analysis of 2 2 systems and nonlinear second order equations... 155 8.1 Planarhamiltoniansystems... 156 8.2 Aprey-predatorsystem... 158 8.2.1 The case of fishing... 163 8.3 Phaseplaneanalysis... 164 8.4 On the equation x 00 D f.x/... 165 8.4.1 A first example: The equation x 00 D x x 3... 166 8.4.2 A second example: The equation x 00 D xcx 3... 168 8.5 Exercises... 170 9 Sturm Liouville eigenvalue theory... 173 9.1 Eigenvaluesandeigenfunctions... 174 9.2 Existenceandpropertiesofeigenvalues... 175 9.3 Anapplicationtotheheatequation... 179 9.4 Exercises... 182 10 Solutions by infinite series and Bessel functions... 183 10.1 Solving second order equations by series... 183 10.2 Brief review of power series...... 183 10.3 Series solutions around ordinary points... 185 10.4 The Frobenius method..... 190 10.5 The Bessel equations....... 193 10.5.1 The Bessel equation of order 0... 195 10.5.2 The Bessel equation of order 1... 198 10.5.3 Bessel equations of order m... 200 10.5.4 Some properties of the Bessel functions...... 201 10.6 Exercises...... 205 11 Laplace transform... 207 11.1 Definition and preliminary examples.... 207 11.2 Properties of the Laplace transform...... 210 11.3 Inverse Laplace transform... 215 11.3.1 Convolution... 219 11.4 Laplace transform and differential equations.... 220 11.5 Generalized solutions...... 222 11.6 Appendix: The Dirac delta... 225 11.7 Exercises...... 230 12 Stability theory... 233 12.1 Definition of stability...... 233 12.2 Liapunov direct method.... 234 12.3 Stability of linear systems and n-th order linear equations.... 236 12.3.1 Stability of 2 2 systems... 236 12.3.2 Stability of n n linearsystems... 242 12.3.3 Stability of n-th order linear equations.... 244
xiv Contents 12.4 Hamiltonian systems....... 245 12.5 Stability of equilibria via linearization... 247 12.5.1 Stable and unstable manifolds.... 249 12.6 An asymptotic result... 251 12.7 Exercises...... 254 13 Boundary value problems... 259 13.1 Boundary value problems for autonomous equations... 259 13.1.1 Examples..... 262 13.2 The Green function... 265 13.3 Sub- and supersolutions.... 269 13.4 A nonlinear eigenvalue problem... 273 13.5 Exercises...... 275 14 ERRATA.................................................... E1 Appendix A. Numerical methods... 277 A.1 Firstorderapproximation:Euler smethod... 278 A.1.1 Improved Euler s method... 279 A.2 The Runge Kutta method.... 282 Answers to selected exercises... 287 References... 301 Index... 303