Titles in the Walter Rudin Student Series in Advanced Mathematics Bona, Miklos Introduction to Enumerative Combinatorics Chartrand, Gary and Ping Zhang Introduction to Graph Theory Davis, Sheldon Topology Dumas, Bob and John E. McCarthy Transition to Higher Mathematics: Structure and Proof Rudin, Walter Functional Analysis, 2nd Edition Rudin, Walter Principles ofMathematical Analysis, 3rd Edition Rudin, Walter Real and Complex Analysis, 3rd Edition Simmons, George F. and Steven G. Krantz Difef rential Equations: Theory, Technique, and Practice Walter Rudin Student Series in Advanced Mathematics-Editorial Board Editor-in-Chief: Steven G. Krantz, Washington University in St. Louis David Barrett Jean-Pierre Rosay University of Michigan University of Wisconsin Steven Bell Jonathan Wahl Purdue University University of North Carolina John P. D'Angelo Lawrence Washington University of Illinois at Urbana-Champaign University of Maryland Robert F efferman C. Eugene Wayne University of Chicago Boston University William McCallum Michael Wolf University of Arizona Rice University Bruce Palka Hung-Hsi Wu University of Texas at Austin University of California, Berkeley Harold R. Parks Oregon State University

Differential Equations Theory, Technique, and Practice George F. Simmons Colorado College and Steven G. Krantz Washington University in St. Louis !R Higher Education Boston Burr Ridge, IL Dubuque, IA Madison, WI New York San Francisco St. Louis Bangkok Bogota Caracas Kuala Lumpur Lisbon London Madrid Mexico City Milan Montreal New Delhi Santiago Seoul Singapore Sydney Taipei Toronto

The McGrow·Hill Com antes • • Higher Education DIFFERENTIAL EQUATIONS: THEORY, TECHNIQUE, AN D PRACTICE Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020. Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning Some ancillaries, including electronic and print components, may not be available to customers outside the United States. This book is printed on acid-free paper. 2 3 4 5 6 7 8 9 0 DOC/DOC 0 9 8 7 6 ISBN-13 978-0-07-286315-4 ISBN-IO 0-07-286315-3 Publisher: Elizabeth J. Haefele Senior Sponsoring Editor: Elizabeth Covello Developmental Editor: Dan Seibert Senior Marketing Manager: Nancy Anselment Bradshaw Project Manager: April R. Southwood Senior Production Supervisor: Sherry L. Kane Designer: Laurie B. Janssen Cover Illustration: Rokusek Design Lead Photo Research Coordinator: Carrie K. Burger Supplement Producer: Melissa M. Leick Compositor: The GTS Companies Typeface: 10112 Times Roman Printer: R. R. Donnelley Crawfordsville, IN Library of Congress Cataloging-in-Publication Data Simmons, George Finlay, 1925-. Differential equations : theory, technique, and practice/George F. Simmons, Steven G. Krantz. - !st ed. p. cm. Includes bibliographical references and index. ISBN 978-0-07-286315-4- ISBN 0-07-286315-3 (acid-free paper) 1. Differential equations-Textbooks. I. Krantz, Steven G. (Steven George), 1951-. II. Title. QA371.S465 2007 2005051118 515 .35-<lc22 CIP www.mhhe.com

PREFACE VIII 1CHAPTER WHAT IS A DIFFERENTIAL EQUATION? 1 2CHAPTER 1.1 Introductory Remarks 2 1.2 The Nature of Solutions 4 1.3 Separable Equations 10 1.4 First-Order Linear Equations 13 1.5 Exact Equations 17 1.6 22 1.7 Orthogonal Trajectories and Families of Curves 26 1.8 29 1.9 Homogeneous Equations 33 Integrating Factors 33 1.10 Reduction of Order 35 1.9.l Dependent Variable Missing 38 1.11 1.9.2 Independent Variable Missing 38 The Hanging Chain and Pursuit Curves 42 1.10.1 The Hanging Chain 45 1.10.2 Pursuit Curves Electrical Circuits 49 Anatomy of an Application: The Design of a 53 Dialysis Machine Problems for Review and Discovery 57 SECOND-ORDER LINEAR EQUATIONS 58 63 2.1 Second-Order Linear Equations with Constant 67 71 Coefficients 75 75 2.2 The Method of Undetermined Coefficients 77 2.3 The Method of Variation of Parameters 80 2.4 The Use of a Known Solution to Find Another 82 2.5 Vibrations and Oscillations iii 2.5.1 Undamped Simple Harmonic Motion 2.5.2 Damped Vibrations 2.5.3 Forced Vibrations 2.5.4 A Few Remarks about Electricity

iv Table of Contents 2.6 Newton's Law of Gravitation and Kepler's Laws 84 2.6.l Kepler's Second Law 86 2.6.2 Kepler's First Law 87 2.6.3 Kepler's Third Law 89 2.7 Higher Order Linear Equations, Coupled 93 Harmonic Oscillators 99 Historical Note: Euler Anatomy of an Application: Bessel 101 Functions and the Vibrating Membrane 105 Problems for Review and Discovery 3CHAPTER QUALITATIVE PROPERTIES AND· THEORETICAL ASPECTS 109 3.1 Review of Linear Algebra 3.1.l Vector Spaces 110 3.2 3.1.2 The Concept of Linear Independence 110 3.3 3.1.3 Bases 111 3.1.4 Inner Product Spaces 113 3.4 3.1.5 Linear Transformations and Matrices 114 3.5 3.1.6 Eigenvalues and Eigenvectors 115 A Bit of Theory 117 Picard's Existence and Uniqueness Theorem 119 3.3.l The Form of a Differential Equation 125 3.3.2 Picard's Iteration Technique 125 3.3.3 Some Illustrative Examples 126 3.3.4 Estimation of the Picard Iterates 127 Oscillations and the Sturm Separation Theorem 129 The Sturm Comparison Theorem 130 Anatomy of an Application: The Green's 138 Function Problems for Review and Discovery 142 146 4CHAPTER POWER SERIES SOLUTIONS AND SPECIAL FUNCTIONS 149 4.1 Introduction and Review of Power Series 4.1.l Review of Power Series 150 150 4.2 Series Solutions of First-Order Differential Equations 159

Table of Contents v 4.3 Second-Order Linear Equations: Ordinary Points 164 4.4 Regular Singular Points 171 4.5 More on Regular Singular Points 177 4.6 Gauss's Hypergeometric Equation 184 189 Historical Note: Gauss 190 Historical Note: Abel Anatomy of an Application: Steady-State 192 Temperature in a Ball 194 Problems for Review and Discovery 5CHAPTER FOURIER SERIES: BASIC CONCEPTS 197 5.1 Fourier Coefficients 198 5.2 Some Remarks about Convergence 207 5.3 Even and Odd Functions: Cosine and Sine Series 211 5.4 Fourier Series on Arbitrary Intervals 218 5.5 Orthogonal Functions 221 225 Historical Note: Riemann Anatomy of an Application: Introduction to 227 the Fourier Transform 236 Problems for Review and Discovery 6CHAPTER PARTIAL DIFFERENTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS 239 6.1 Introduction and Historical Remarks 6.2 Eigenvalues, Eigenfunctions, and the Vibrating 240 String 243 6.2.1 Boundary Value Problems 243 6.2.2 Derivation of the Wave Equation 244 6.2.3 Solution of the Wave Equation 246 6.3 The Heat Equation 251 6.4 The Dirichlet Problem for a Disc 256 6.4.1 The Poisson Integral 259 6.5 Sturm-Liouville Problems 262 Historical Note: Fourier 267 Historical Note: Dirichlet 268 Anatomy of an Application: Some Ideas from Quantum Mechanics 270 Problems for Review and Discovery 273

vi Table of Contents 7CHAPTER LAPLACE TRANSFORMS 277 8CHAPTER 9CHAPTER 7.1 Introduction 278 10CHAPTER 7.2 Applications to Differential Equations 280 7.3 Derivatives and Integrals of Laplace Transforms 285 7.4 Convolutions 291 293 7.4.1 Abel's Mechanical Problem 298 7.5 The Unit Step and Impulse Functions 305 Historical Note: Laplace 306 Anatomy of an Application: Flow Initiated by 309 an Impulsively Started Flat Plate · Problems for Review and Discovery THE CALCULUS OF VARIATIONS 315 8.1 Introductory Remarks 316 8.2 Euler's Equation 319 8.3 Isoperimetric Problems and the Like 327 328 8.3.1 Lagrange Multipliers 329 8.3.2 Integral Side Conditions 333 8.3.3 Finite Side Conditions 338 Historical Note: Newton Anatomy of an Application: Hamilton's 340 Principle and its Implications 344 Problems for Review and Discovery NUMERICAL METHODS 347 9.1 Introductory Remarks 348 9.2 The Method of Euler 349 9.3 The Error Term 353 9.4 An Improved Euler Method 357 9.5 The Runge-Kutta Method 360 Anatomy of an Application: A Constant 365 Perturbation Method for Linear, 368 Second-Order Equations Problems for Review and Discovery SYSTEMS OF FIRST-ORDER 371 EQUATIONS 372 10.l Introductory Remarks 374 10.2 Linear Systems

Table of Contents vii 10.3 Homogeneous Linear Systems with Constant 382 10.4 Coefficients 389 Nonlinear Systems: Volterra's Predator-Prey 395 Equations 400 Anatomy of an Application: Solution of Systems with Matrices and Exponentials Problems for Review and Discovery 11CHAPTER THE NONLINEAR THEORY 403 12CHAPTER 11.1 Some Motivating Examples 404 11.2 Specializing Down 404 11.3 Types of Critical Points: Stability 409 11.4 Critical Points and Stability for Linear Systems 417 11.5 Stability by Liapunov's Direct Method 427 11.6 Simple Critical Points of Nonlinear Systems 432 11.7 Nonlinear Mechanics: Conservative Systems 439 11.8 Periodic Solutions: The Poincare-Bendixson Theorem 444 Historical Note: Poincare 452 Anatomy of an Application: Mechanical Analysis of a Block on a Spring 454 Problems for Review and Discovery 457 DYNAMICAL SYSTEMS 461 12.1 Flows Dynamical Systems 462 12.2 12.1.1 Stable and Unstable Fixed Points 464 12.3 12.1.2 Linear Dynamics in the Plane 466 12.1.3 468 475 Some Ideas from Topology 475 12.2.1 Open and Closed Sets 476 12.2.2 The Idea of Connectedness 478 12.2.3 Closed Curves in the Plane 480 480 Planar Autonomous Systems 12.3.1 Ingredients of the Proof of Poincare-Bendixson 489 493 Anatomy of an Application: Lagrange's Equations 495 497 Problems for Review and Discovery 525 BIB LIOG RAP HY ANSWERS TO ODD-NUMBERED EXERCISES INDEX

Differential equations is one of the oldest subjects in modem mathematics. It was not Jong after Newton and Leibniz invented calculus that Bernoulli and Euler and others began to consider the heat equation and the wave equation of mathematical physics. Newton himself solved differential equations both in the study of planetary motion and also in his consideration of optics. Today differential equations is the centerpiece of much of engineering, of physics, of signif icant parts of the life sciences, and in many areas of mathematical modeling. The audience for a sophomore course in ordinary differential equations is substantial-second only perhaps to that for calculus. There is a need for a definitive text that both describes classical ideas and provides an entree to the newer ones. Such a text should pay careful attention to advan_ced topics like the Laplace transform, Sturm-Liouville theory, and boundary value problems (on the traditional side) but should also pay due homage to nonlinear theory, to dynamics, to modeling, and to computing (on the modem side). George Simmons's fine text is a traditional book written in the classical style. It provides a cogent and accessible introduction to all the traditional topics. It is a pleasure to have this opportunity to bring this text up to date and to add some more timely material. We have streamlined some of the exposition and augmented other parts. There is now computer work based not only on number crunching but also on computer algebra systems such as Maple, Mathematica, and .MATLAB. Certainly a study of flows and vector fields, and of the beautiful Poincare-Bendixson theory built thereon, is essential for any modem treatment. One can introduce some of the modem ideas from the theory of dynamics to obtain qualitative information about nonlinear differential equations and systems. And all of the above is a basis for modeling. Modeling is what brings the subject to life and makes the ideas real for the students. Differential equations can model real-life questions, and computer calculations and graphics can then provide real-life answers. The symbiosis of the synthetic and the calculational provides a rich educational experience for students, and it prepares them for more concrete, applied work in future courses. The new Anatomy of an Application sections in this edition showcase some rich applications from engineering, physics, and applied science. There are a number of good ordinary differential equations books available today. Popu lar standards include Boyce & DiPrima; Nagle, Saff, & Snider; Edwards & Penney; Derrick & Grossman; and Polking, Boggess & Arnold. Books for a more specialized audience include Amol'd; Hubbard & Hubbard; Borrelli & Coleman; and Blanchard, Devaney, & Hall. Classical books, still in use at some schools, include Coddington & Levinson and Birkhoff & Rota. Each of these books has some strengths, but not the combination of features that we have planned for Simmons & Krantz. None has the crystal clear and elegant quality of writing for which George Simmons is so well known. Steven G. Krantz is also a mathematical writer of some repute (50 books and 140 papers), and can sustain the model set by Simmons in this new edition. No book will have the well-developed treatment of modeling and computing (done in a manner so that these two activities speak to each other) that will be rendered in Simmons & Krantz. None will have the quality of exercises. We look forward to setting a new standard for the modem textbook on ordinary differential equations, a standard to which other texts may aspire. This will be a book that students read, and internalize, and in the end apply to other subjects and disciplines. It will Jay the foundation for

Preface ix future studies in analytical thinking. It will be a touchstone for in-depth studies of growth and change. Key Features • Anatomy of An Application - Occurring at the end of each chapter, these in-depth exami nations of particular applications of ordinary differential equations motivate students to use critical thinking skills to solve practical problems in engineering, physics, and the sciences. After the application is introduced in context, the key concepts and procedures needed to model its associated problems are presented and discussed in detail. • Exercises - The text contains a wide variety of section-level exercises covering varying levels of difficulty. Hints are given when appropriate to assist students with difficult prob lems and crucial concepts. Special technology exercises are included in nearly every section which harness the power of computer algebra systems such as Maple, Mathematica, and MATLAB for solving ordinary differential equations. Answers to the odd-numbered exercises in the text are included in the Answers to Odd-Numbered Exercises at the back of the book. • Problems for Review and Discovery - Each chapter is concluded with three sets of review exercises. Drill Exercises test students' basic understanding of key concepts from the chapter. Challenge Problems take that review a step further by presenting students with more complex problems requiring a greater degree of critical thinking. And Problems for Discussion and Exploration offer students open-ended opportunities to explore topics from the chapter and develop their intuition and command of the material. • Historical Notes - These biographies, occurring at the end of chapters, offer fascinating in sight into the lives and accomplishments of some of the great mathematicians who contributed to the development of differential equations. A longtime hallmark of George Simmons' writ ings, the Historical Notes show how mathematics is at its heart a human endeavor developed to meet human needs. • Math Nuggets - These brief asides, appearing throughout the text, offer quick historical context and interesting anecdotes tied to the specific topic under discussion. They serve to underscore the human element behind the development of ordinary differential equ<1tions in shorter and more context-sensitive form than the end-of-chapter Historical Notes. Supplements Student's Solutions Manual, by Donald Hartig (ISBN-JO: 0-07-286316-1, ISBN-13: 978-0-07- 286316-1) - Contains complete worked solutions to odd-numbered exercises from the text. Instructor's Solutions Manual, by Donald Hartig (ISBN-10: 0-07-323091-X, ISBN-13: 978-0- 07-323091-7) - Contains complete worked solutions to even-numbered exercises from the text. Companion Website, http://www.mhhe.com/simmons - Contains free online resources for stu dents and instructors to accompany the text. The website features online technology manuals for computer algebra systems such as Maple and Mathematica. These technology manuals give a general overview of these systems and how to use them to solve and explore ordinary differen tial equations, and provide additional problems and worksheets for further practice with these computational tools.

x Preface Acknowledgements We would like to thank the following individuals who reviewed the manuscript and provided valuable suggestions for improvement: Yuri Antipov, Louisiana State University Dieter Armbruster, Arizona State University Vitaly Bergelson, Ohio State University James Ward Brown, University of Michigan - Dearborn Nguyen Cac, University of Iowa Benito Chen, University of Wyoming Goong Chen, Texas A&M University Ben Cox, College of Charleston Richard Crew, University of Florida Moses Glasner, Pennsylvania State University David Grant, University of Colorado Johnny Henderson, Baylor University Michael Kirby, Colorado State University Przemo Kranz, University of Mississippi Melvin Lax, California State University - Long Beach William Margulies, California State University - Long Beach James Okon, California State University - San Bernardino William Paulsen, Arkansas State University Jonathan Rosenberg, University of Maryland Jairo Santanilla, University of New Orleans Michael Shearer, North Carolina State University Jie Shen, Purdue University Marshall Slemrod, University of Wisconsin P.K. Subramanian, California State University - Los Angeles Kirk Tolman, Brigham Young University Xiaoming Wang, Florida State University Steve Zelditch, Johns Hopkins University Zhengfang Zhou, Michigan State University Special thanks go to Steven Boettcher, who prepared the answer key; Donald Hartig, who prepared the two solutions manuals; and Daniel Zwillinger, who checked the complete text and exercises for accuracy. T he previous incarnation of this classic text was written by George F. Simmons. His book has served as an inspiration to several generations of differential equations students. It has been a pleasure to prepare this new version for a new body of students. George Simmons has played a proactive role at every stage of the writing process, contributing many ideas, edits, and corrections. His wisdom pervades the entire text. Steven G. Krantz For Hope and Nancy my wife and daughter who still make it all worthwhile For Randi and Hypatia my wife and daughter who know why I dedicate my books to them

1CHAPTER --------· ---- -Wfi�t Is a Differential Equation? • The concept of a differential equation • Characteristics of a solution • Finding a solution • Separable equations • First-order linear equations • Exact equations • Orthogonal trajectories

2 Chapter 1 What Is a Differential Equation? ,.. INTRODUCTORY REMARKS A differential equation is an equation relating some function f to one or more of its derivatives. An example is d2f (x) + 2x-ddfx (x) +f2 (x) = sin x. (1.1) d-x 2 Observe that this particular equation involves a function f together with its first and second derivatives. Any given differential equation may or may not involve f or any particular derivative off. But, for an equation to be a differential equation, at least some derivative off must appear. The objective in solving an equation like Equation (1.1) is tofind thefunctionf. Thus we already perceive a fundamental new paradigm: When we solve an algebraic equation, we seek a number or perhaps a collection of numbers; but when we solve a differential equation we seek one or more functions. Many of the laws of nature-in physics, in chemistry, in biology, in engineering, and in astronomy-find their most natural expression in the language of differential equa tions. Put in other words, differential equations are the language of nature. Applications of differential equations also abound in mathematics itself, especially in geometry and harmonic analysis and modeling. Differential equations occur in economics and systems science and other fields of mathematical science. It is not difficult to perceive why differential equations arise so readily in the sciences. If y = f(x) is a given function, then the derivativedf/dx can be interpreted as the rate of change off with respect to x. In any process of nature, the variables involved are related to their rates of change by the basic scientific principles that govern the process-that is, by the laws of nature. When this relationship is expressed in mathematical notation, the result is usually a differential equation. Certainly Newton's law of universal gravitation, Maxwell's field equations, the motions of the planets, and the refraction of light are important examples which can be expressed using differential equations. Much of our understanding of nature comes from our ability to solve differential equations. The purpose of this book is to introduce you to some of these techniques. The following example will illustrate some of these ideas. According to Newton's second law of motion, the acceleration a of a body (of massm) is proportional to the total force F acting on the body. The standard expression of this relationship is F =m · a . (1.2) Supopes in particular that we are analyzing a falling body. Express the height of the body from the surface of the Earth as y(t) feet at time t. The only force acting on the body is that due to gravity. If g is the acceleration due to gravity (about -32 ft/sec2 near the surface of the Earth) then the force exerted on the body ism · g. And of course the acceleration is d2y/dt2• Thus Newton's law, Equation (1.2), becomes (1.3)

Section 1.1 Introductory Remarks 3 or d2y g= dt2 • We may make the problem a little more interesting by supposing that air exerts a - (1.3)the total force acting on the body is mg k·(dy/dt). Then Equation resisting force proportional to the velocity. If the constant of proportionality is k, then -m·g becomes - k·-dy = d2y . (1.4) m· dt dt2 Equations (1.3) and (1.4) express the essential attributes of this physical system. A few additional examples of differential equations are - d2y dy (1 x2) -2x +p(p +l)y = 0; (1.5) (1.6) dx2 dx -d2y dy x2 +x- + {x2 - p2) y = O; dx2 dx d2y = O; (1.7) +xy dx2 (1-x2)y\"-xy' +p2y = O; (1.8) y\" - 2xy' + 2py = 0; (1.9) -ddxy =k·y; (1.10) (1.11) ( )d3y dy 2 + = y 3 +sinx. dx3 dx Equations (l.5)-(1.9) are called Legendre's equation, Bessel's equation, Airy's equation, Chebyshev's equation, and Hermite's equation, respectively. Each has a vast literature and a history reaching back hundreds of years. We shall touch on each of these equations (1.10)later in the book. Equation is the equation of exponential decay (or of biological growth). Adrien Marie Legendre (1752-1833) invented Legendre polynomials (the contri bution for which he is best remembered) in the context of gravitational attraction of ellipsoids. Legendre was a fine French mathematician who suffered the misfor tune of seeing most of his best work-in elliptic integrals, number theory, and the method of least squares-superseded by the achievements of younger and abler men. For instance, he devoted 40 years to the study of elliptic integrals, and his

4 Chapter 1 What Is a Differential Equation? two-volume treatise on the subject had scarcely appeared in print before the dis coveries of Abel and Jacobi revolutionized the field. Legendre was remarkable for the generous spirit with which he repeatedly welcomed newer and better work that made his own obsolete. Each of Equations ( l .5)-( 1.9) is of second order, meaning that the highest derivative that appears is the second. Equation (l. 1 0) is of first order. Equation (1. 1 l) is of third order.Also, each of Equations ( l .5)-( l. l0) is linear, while ( l. l l) is nonlinear.We shall say more about this distinction below.Each equation is an ordinary difef rential equation, meaning that it involves a function of a single variable and the (Jrdinary derivatives of that function. A partial differential equation is one involving a function of two or more variables, and in which the derivatives are partial derivatives. These equations are more subtle, and more difficult, than ordinary differential equations. We shall say something about partial differential equations near the end of the book. Friedrich Wilhelm Bessel (1784-1846) was a distinguished German astronomer and an intimate friend of Gauss. The two corresponded for many years. Bessel was the first man to determine accurately the distance of a fixed star (the star 61 Cygni). In 1844 he discovered the binary (or twin) star Sirius. The companion star to Sirius has the size of a planet but the mass of a star; its density is many thousands of times the density of water. It was the first dead star to be discovered, and it occupies a special place in the modem theory of stellar evolution. .f'.t-THE NATURE OF SOLUTIONS An ordinary differential equation of order n is an equation involving an unknown function f together with its derivatives df d2f dnf dx ' dx2 ' • • · ' dxn We might, in a more formal manner, express such an equation as , ��)( �;, �7i,F x, f, ... = 0. How do we verify that a given function f is actually the solution of such an equation? The answer to this question is best understood in the context of concrete examples. Note that we often denote the unknown function by y, as in y = f(x).

Section 1.2 The Nature of Solutions 5 EXAMPLE 1.1 Consider the differential equation y\" -Sy'+ 6y = 0. Without saying how the solutions are actuallyfound, verify that y1(x) = e2x and y2(x) = e3x are both solutions. Solution To verify this assertion, we note that • Y1\" -SY1t + 6Y1 = 2 · 2 · e2x S- · 2 · e2x + 6 · e 2x = [4 - 10+ 6] · e2x =0 and y{ -Sy�+ 6y2 = 3 · 3 · e3x - S · 3 · e3x + 6 · e3x = [9 - lS + 6] · e3x ::0. This process, of verifying that a function is a solution of the given differential equation, is entirely new. The reader will want to practice and become accustomed to it. In the present instance, the reader may check that any function of the form (1 .12) (where c1, c2 are arbitrary constants) is also a solution of the differential equation in Example 1.1. An important obverse consideration is this: When you are going through the pro cedure to solve a differential equation, how do you know when you are finished? The answer is that the solution process is complete when all derivatives have been eliminated from the equation. For then you will have y expressed in terms of x, at least implicitly. Thus you will have found the sought-after function. For a large class of equations that we shall study in detail in the present book, we shall find a number of \"independent\" solutions equal to the order of the differential equation. Then we shall be able to form a so-called general solution by combining them as in Equation (1.12). Picard's Existence and Uniqueness Theorem, covered in detail in Section 3.2, will help us to see that our general solution is complete-there are no other solutions. Of course we shall provide all the details in the development below. Sometimes the solution of a differential equation will be expressed as an implicitly defined function. An example is the equation dy y2 (1.13) dx = 1 -xy'

6 Chapter 1 What Is a Differential Equation? which has solution xy= lny+c . (1.14) Note here that the hallmark of what we call a solution is that it has no derivatives in it: It is a direct formula, relating y (the dependent variable) to x (the independent variable). To verify that Equation (1.14) is indeed a solution of Equation (1.13), let us differentiate: dd -[xy] = -[lny+c] dx dx hence --dy dy/dx l·y+x·-= dx y or (� )dy -x = y. dx y In conclusion, -dy y2 = dx 1 xy' as desired. Note that it would be difficult to solve (l.14) for y in terms of x. One unifying feature of the two examples that we have now seen of verifying solutions is this: When we solve an equation of order n, we expect n \"independent solutions\" (we shall have to say later just what this word \"independent\" means) and we expect n undetermined constants. In the first example, the equation was of order 2 and the undetermined constants were c1 and c2. In the second example, the equation was of order 1 and the undetermined constant was c. Sir George Biddell Airy (1801-1892) was Astronomer Royal of England for many years. He was a hard-working, systematic plodder whose sense of decorum almost deprived John Couch Adams of credit for discovering the planet Neptune. As a boy, Airy was notorious for his skill in designing peashooters. Although this may have been considered to be a notable start, and in spite of his later contributions to the theory of light (he was one of the first to identify the medical condition known as astigmatism), Airy seems to have developed into an excessively practical sort of scientist who was obsessed with elaborate numerical calculations. He had little use for abstract scientific ideas. Nonetheless, Airy functions still play a prominent role in differential equations, special function theory, and mathematical physics.

Section 1.2 The Nature of Solutions 7 EXAMPLE 1.2 Verify that, for any choice of the constants A and B, the function y =x2 + Aex + Be-x is a solution of the differential equation y\" - y =2 - x2• Solution This solution set is typical of what we shall learn to find for a second-order linear equation. There are two free parameters in the solution (corresponding to the degree two of the equation). Now, if y =x2 + Aex + Be-x, then y' =2x + Aex - Be-x and y\" =2 + Aex + Be-x. Hence y\" - y = [2 + Aex + ]Be-x - [x2 + Aex + ]Be-x =2 - x2 as required. • Remark 1.1 One of the powerful and fascinating features of the study of differential equations is the geometric interpretation that we can often place on the solution of a problem. This connection is most apparent when we consider a first-order equation. Consider the equation -dy = F(x, y). (1.15) dx We may think of the Equation (l.15) as assigning to each point (x, y) in the plane a slope dy/dx. For the purposes of drawing a picture, it is more convenient to think of the equation as assigning to the point (x, y) the vector (1, dy/dx). See Figure 1.1. Figure 1.2 illustrates how the differential equation -ddxy =x assigns such a vector to each point in the plane. Figure 1.3 illustrates how the differential equation -ddxy =-y assigns such a vector to each point in the plane. Chapter 11 will explore in greater detail how the geometric analysis of these so called vector fields can lead to an understanding of the solutions of a variety of differential equations. •

8 Chapter 1 What Is a Differential Equation? V }(x. dy/dx � I FIGURE 1.1 3x FIGURE 1.2 y 23 FIGURE 1.3

Section 1.2 The Nature of Solutions 9 EXERCISES 1. Verify that the following functions (explicit or implicit) are solutions of the corresponding differential equations: (a) y=x2+c y'=2x (b) y=cx2 xy'=2y (c) y2=e2x+c yy'=e2x (d) y=cekx y'=ky (e) yy== CCttes2inx+2xC+2e-c22xcos 2x y\"+4y=0 (f) y\"-4y=0 (g) y=Ctsinh 2x+c2 cosh 2x y\" -4y=0 (h) y=arcsin xy xy'+y=y'Jl-x2y xy'=y+x2+y2 (i) y=xtan x yI= XY (j) x2=2y2 ln y x2+y2 2xyy'=x2+y2 (k) y2=x2-ex y+xy'=x4(y')2 o) y=c2+cIx y'=y2 /(xy-x2) (m) y=ceyfx (ycos y - sin y+x)y'=y (n) y+sin y=x 1+y2+y2y'=0 (o) x+y=arctan y 2. Find the general solution of each of the following differential equations: (a) y'=e3x-x (f) xy'=1 (b) y'=xex2 (g) yI=arcs •mx (c) (1+x)y'=x (h) y'sin x=1 (d) (1+x2)y'=x (i) (1+x3)y'=x (e) (1+x2)y'=arct an x (j) (x2-3x+2)y'=x 3. For each of the following differentia l equations, find the particular solution that satisfies the given initial condition: (a) y'=xex y=3 when x=1 (b) y'=2sin xcos x y=1 when x=0 (c) y'=ln x y=Owhen x=e (d) (x2-l)y'=1 y=Owhen x=2 (e) x(x2-4)y' = 1 y=0when x=1 (f) (x+l)(x2+l)y'=2x2+x y=1 when x=0 4. Show that the function iy=ex2 x e-12 dt is a solution of the differential equation y'=2xy+1. 5. For the differential equation y\"-5y'+4y=0, carry out the detailed calculations required to verify these assertions: (a) The functions y=ex and y=e4x are both solutions. (b) The function y=Ctex+c2e4x is a solution for any choice ofconstants Ct, c2.

10 Chapter 1 What Is a Differential Equation? 6. Verify that x2 y = In y + c is a solution of the differential equation dy/dx = ( )2xy2 / 1 - x2 y for any choice of the constant c. 7. For which values of m will the function y = Ym = emx be a solution of the differential equation 2y111 + y\" - Sy' + 2y = 0? Find three such values m. Use the ideas in Exercise 5 to find a solution containing three arbitrary constants c1 ,.c2, c3. EPARABLE EQUATIONS In this section we shall encounter our first general class of equations with the property that (i) We can immediately recognize members of this class of equations. (ii) We have a simple and direct method for (in principle) solving such equations. This is the class ofseparable equations. A first-order ordinary differential equation isseparable if it is possible, by elemen tary algebraic manipulation, to arrange the equation so that all the dependent variables (usually the y variable) are on one side and all the independent variables (usually the x variable) are on the other side. Let us learn the method by way of some examples. EXAMPLE 1.3 Solve the ordinary differential equation y' = 2xy. Solution In the method of separation of variables-which is a method for first-order equations only-it is useful to write the derivative using Leibniz notation. Thus we have dy - = 2xy. dx We rearrange this equation as dy - = 2xdx . y [It should be noted here that we use the shorthanddy to stand for (dy/dx)-dx and we of course assume that y =f. 0.] Now we can integrate both sides of the last displayed equation to obtain f : fd = 2xdx. We are fortunate in that both integrals are easily evaluated. We obtain In IYI = x2 + c .

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