华章数学译丛 微分方程与边界值问题 英文.pdf

Licensed to: iChapters User Differential Equations with Boundary-Value Problems, Seventh Edition Dennis G. Zill and Michael R. Cullen Executive Editor: Charlie Van Wagner Development Editor: Leslie Lahr Assistant Editor: Stacy Green Editorial Assistant: Cynthia Ashton Technology Project Manager: Sam Subity Marketing Specialist: Ashley Pickering Marketing Communications Manager: Darlene Amidon-Brent Project Manager, Editorial Production: Cheryll Linthicum Creative Director: Rob Hugel Art Director: Vernon Boes Print Buyer: Rebecca Cross Permissions Editor: Mardell Glinski Schultz Production Service: Hearthside Publishing Services Text Designer: Diane Beasley Photo Researcher: Don Schlotman Copy Editor: Barbara Willette Illustrator: Jade Myers, Matrix Cover Designer: Larry Didona Cover Image: Getty Images Compositor: ICC Macmillan Inc. 2009, 2005 Brooks/Cole, Cengage Learning ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher. Library of Congress Control Number: 2008924835 ISBN-13: 978-0-495-10836-8 ISBN-10: 0-495-10836-7 Brooks/Cole 10 Davis Drive Belmont, CA 94002-3098 USA Cengage Learning is a leading provider of customized learning solutions with offi ce locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan. Locate your local offi ce at Cengage Learning products are represented in Canada by Nelson Education, Ltd. For your course and learning solutions, visit . Purchase any of our products at your local college store or at our preferred online store . For product information and technology assistance, contact us at Cengage Learning Customer the fourth derivative is written y(4)instead of y?. In general, the nth derivative of y is written dny?dxnor y(n). Although less convenient to write and to typeset, the Leibniz notation has an advan- tage over the prime notation in that it clearly displays both the dependent and independent variables. For example, in the equation it is immediately seen that the symbol x now represents a dependent variable, whereas the independent variable is t. You should also be aware that in physical sciences and engineering, Newtons dot notation (derogatively referred to by some as the “fl yspeck” notation) is sometimes used to denote derivatives with respect to time t. Thus the differential equation d2s?dt2? ?32 becomes s ? ?32. Partial derivatives are often denoted by a subscript notation indicating the indepen- dent variables. For example, with the subscript notation the second equation in (3) becomes uxx? utt? 2ut. CLASSIFICATION BY ORDERThe order of a differential equation (either ODE or PDE) is the order of the highest derivative in the equation. For example, is a second-order ordinary differential equation. First-order ordinary differential equations are occasionally written in differential form M(x, y) dx ? N(x, y) dy ? 0. For example, if we assume that ydenotes the dependent variable in (y ? x) dx ? 4xdy ? 0, then y? ? dy?dx, so by dividing by the differential dx, we get the alternative form 4xy? ? y ? x. See the Remarks at the end of this section. In symbols we can express an nth-order ordinary differential equation in one dependent variable by the general form ,(4) where F is a real-valued function of n ? 2 variables: x, y, y?, . . . , y(n). For both prac- tical and theoretical reasons we shall also make the assumption hereafter that it is possible to solve an ordinary differential equation in the form (4) uniquely for the F(x, y, y?, . . . , y(n) ? 0 first ordersecond order ? 5( ) 3 ? 4y ? ex dy dx d2y dx2 d2x dt2 ? 16x ? 0 unknown function or dependent variable independent variable ?2u ?x2 ? ?2u ?y2 ? 0, ?2u ?x2 ? ?2u ?t2 ? 2?u ?t, and ?u ?y ? ?v ?x 1.1DEFINITIONS AND TERMINOLOGY 3 *Except for this introductory section, only ordinary differential equations are considered in A First Course in Differential Equations with Modeling Applications, Ninth Edition. In that text the word equation and the abbreviation DE refer only to ODEs. Partial differential equations or PDEs are considered in the expanded volume Differential Equations with Boundary-Value Problems, Seventh Edition. 08367_01_ch01_p001-033.qxd 4/7/08 1:04 PM Page 3 Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Licensed to: iChapters User highest derivative y(n)in terms of the remaining n ? 1 variables. The differential equation ,(5) where f is a real-valued continuous function, is referred to as the normal form of (4). Thus when it suits our purposes, we shall use the normal forms to represent general fi rst- and second-order ordinary differential equations. For example, the normal form of the fi rst-order equation 4xy? ? y ? xis y? ? (x ? y)?4x; the normal form of the second-order equation y? ? y? ? 6y ? 0 is y? ? y? ? 6y. See the Remarks. CLASSIFICATION BY LINEARITYAn nth-order ordinary differential equation (4) is said to be linear if F is linear in y, y?, . . . , y(n). This means that an nth-order ODE is linear when (4) is an(x)y(n)? an?1(x)y(n?1)? ? ? ? ? a1(x)y? ? a0(x)y ? g(x) ? 0 or .(6) Two important special cases of (6) are linear fi rst-order (n ? 1) and linear second- order (n ? 2) DEs: .(7) In the additive combination on the left-hand side of equation (6) we see that the char- acteristic two properties of a linear ODE are as follows: The dependent variable y and all its derivatives y?, y?, . . . , y(n)are of the fi rst degree, that is, the power of each term involving y is 1. The coeffi cients a0, a1, . . . , anof y, y?, . . . , y(n)depend at most on the independent variable x. The equations are, in turn, linear fi rst-, second-, and third-order ordinary differential equations. We have just demonstrated that the fi rst equation is linear in the variable y by writing it in the alternative form 4xy? ? y ? x. A nonlinear ordinary differential equation is sim- ply one that is not linear. Nonlinear functions of the dependent variable or its deriva- tives, such as sin y or , cannot appear in a linear equation. Therefore are examples of nonlinear fi rst-, second-, and fourth-order ordinary differential equa- tions, respectively. SOLUTIONSAs was stated before, one of the goals in this course is to solve, or fi nd solutions of, differential equations. In the next defi nition we consider the con- cept of a solution of an ordinary differential equation. nonlinear term: coeffi cient depends on y nonlinear term: nonlinear function of y nonlinear term: power not 1 (1 ? y)y? ? 2y ? ex,? sin y ? 0,and d2y dx2 ? y2 ? 0 d4y dx4 ey? (y ? x)dx ? 4x dy ? 0, y? ? 2y? ? y ? 0, and d3y dx3 ? x dy dx ? 5y ? ex a1(x) dy dx ? a0(x)y ? g(x) and a2(x) d 2y dx2 ? a1(x) dy dx ? a0(x)y ? g(x) an(x) d ny dxn ? an?1(x) d n?1y dxn?1 ? ? ? ? ? a1(x) dy dx ? a0(x)y ? g(x) dy dx ? f(x, y) and d2y dx2 ? f(x, y, y?) dny dxn ? f(x, y, y?, . . . , y(n?1) 4 CHAPTER 1INTRODUCTION TO DIFFERENTIAL EQUATIONS 08367_01_ch01_p001-033.qxd 4/7/08 1:04 PM Page 4 Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Licensed to: iChapters User DEFINITION 1.1.2Solution of an ODE Any function ?, defi ned on an interval I and possessing at least n derivatives that are continuous on I, which when substituted into an nth-order ordinary differential equation reduces the equation to an identity, is said to be a solution of the equation on the interval. In other words, a solution of an nth-order ordinary differential equation (4) is a func- tion ? that possesses at least n derivatives and for which We say that ? satisfi es the differential equation on I. For our purposes we shall also assume that a solution ? is a real-valued function. In our introductory discussion we saw that is a solution of dy?dx ? 0.2xy on the interval (?, ?). Occasionally, it will be convenient to denote a solution by the alternative symbol y(x). INTERVAL OF DEFINITIONYou cannot think solution of an ordinary differential equation without simultaneously thinking interval. The interval I in Defi nition 1.1.2 is variously called the interval of defi nition, the interval of existence, the interval of validity, or the domain of the solution and can be an open interval (a, b), a closed interval a, b, an infi nite interval (a, ?), and so on. EXAMPLE 1 Verifi cation of a Solution Verify that the indicated function is a solution of the given differential equation on the interval (?, ?). (a)(b) SOLUTIONOne way of verifying that the given function is a solution is to see, after substituting, whether each side of the equation is the same for every x in the interval. (a) From we see that each side of the equation is the same for every real number x. Note that is, by defi nition, the nonnegative square root of . (b) From the derivatives y? ? xex? exand y? ? xex? 2exwe have, for every real number x, Note, too, that in Example 1 each differential equation possesses the constant so- lution y ? 0, ? ? x ? ?. A solution of a differential equation that is identically zero on an interval I is said to be a trivial solution. SOLUTION CURVEThe graph of a solution ? of an ODE is called a solution curve. Since ? is a differentiable function, it is continuous on its interval I of defi ni- tion. Thus there may be a difference between the graph of the function ? and the right-hand side: 0. left-hand side: y? ? 2y? ? y ? (xex? 2ex) ? 2(xex? ex) ? xex? 0, 1 16 x 4 y1/2? 1 4 x 2 right-hand side: xy1/2? x ? 1 16 x 4? 1/2 ? x ?1 4 x 2? ? 1 4 x 3, left-hand side: dy dx ? 1 16 (4 ? x 3) ? 1 4 x 3, y? ? 2y? ? y ? 0; y ? xexdydx ? xy1/2; y ? 1 16 x 4 y ? e0.1x 2 F(x, ?(x), ?(x), . . . , ?(n)(x) ? 0 for all x in I. 1.1DEFINITIONS AND TERMINOLOGY 5 08367_01_ch01_p001-033.qxd 4/7/08 1:04 PM Page 5 Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Licensed to: iChapters User graph of the solution ?. Put another way, the domain of the function ? need not be the same as the interval I of defi nition (or domain) of the solution ?. Example 2 illustrates the difference. EXAMPLE 2Function versus Solution The domain of y ? 1?x, considered simply as a function, is the set of all real num- bers x except 0. When we graph y ? 1?x, we plot points in the xy-plane corre- sponding to a judicious sampling of numbers taken from its domain. The rational function y ? 1?x is discontinuous at 0, and its graph, in a neighborhood of the ori- gin, is given in Figure 1.1.1(a). The function y ? 1?x is not differentiable at x ? 0, since the y-axis (whose equation is x ? 0) is a vertical asymptote of the graph. Now y ? 1?x is also a solution of the linear fi rst-order differential equation xy? ? y ? 0. (Verify.) But when we say that y ? 1?x is a solution of this DE, we mean that it is a function defi ned on an interval I on which it is differentiable and satisfi es the equation. In other words, y ? 1?x is a solution of the DE on any inter- val that does not contain 0, such as (?3, ?1), , (?, 0), or (0, ?). Because the solution curves defi ned by y ? 1?x for ?3 ? x ? ?1 and are sim- ply segments, or pieces, of the solution curves defi ned by y ? 1?x for ? ? x ? 0 and 0 ? x ? ?, respectively, it makes sense to take the interval I to be as large as possible. Thus we take I to be either (?, 0) or (0, ?). The solution curve on (0, ?) is shown in Figure 1.1.1(b). EXPLICIT AND IMPLICIT SOLUTIONSYou should be familiar with the terms explicit functions and implicit functions from your study of calculus. A solution in which the dependent variable is expressed solely in terms of the independent variable and constants is said to be an explicit solution. For our purposes, let us think of an explicit solution as an explicit formula y ? ?(x) that we can manipulate, evaluate, and differentiate using the standard rules. We have just seen in the last two examples that , y ? xex, and y ? 1?x are, in turn, explicit solutions of dy?dx ? xy1/2, y? ? 2y? ? y ? 0, and xy? ? y ? 0. Moreover, the trivial solu- tion y ? 0 is an explicit solution of all three equations. When we get down to the business of actually solving some ordinary differential equations, you will see that methods of solution do not always lead directly to an explicit solution y ? ?(x). This is particularly true when we attempt to solve nonlinear fi rst-order differential equations. Often we have to be content with a relation or expression G(x, y) ? 0 that defi nes a solution ? implicitly. DEFINITION 1.1.3Implicit Solution of an ODE A relation G(x, y) ? 0 is said to be an implicit solution of an ordinary differential equation (4) on an interval I, provided that there exists at least one function ? that satisfi es the relation as well as the differential equation on I. It is beyond the scope of this course to investigate the conditions under which a relation G(x, y) ? 0 defi nes a differentiable function ?. So we shall assume that if the formal implementation of a method of solution leads to a relation G(x, y) ? 0, then there exists at least one function ? that satisfi es both the relation (that is, G(x, ?(x) ? 0) and the differential equation on an interval I. If the implicit solution G(x, y) ? 0 is fairly simple, we may be able to solve for y in terms of x and obtain one or more explicit solutions. See the Remarks. y ? 1 16 x 4 1 2 ? x ? 10 (1 2, 10) 6 CHAPTER 1INTRODUCTION TO DIFFERENTIAL EQUATIONS 1 x y 1 (a) function y ? 1/x, x ? 0 (b) solution y ? 1/x, (0, ?) 1 x y 1 FIGURE 1.1.1The function y ? 1?x is not the same as the solution y ? 1?x 08367_01_ch01_p001-033.qxd 4/7/08 1:04 PM Page 6 Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Licensed to: iChapters User EXAMPLE 3 Verifi cation of an Implicit Solution The relation x2? y2? 25 is an implicit solution of the differential equation (8) on the open interval (?5, 5). By implicit differentiation we obtain . Solving the last equation for the symbol dy?dx gives (8). Moreover, solving x2? y2? 25 for y in terms of x yields . The two functions and satisfy the relation (that is, x2? ?1 2 ? 25 and x2? ?2 2 ? 25) and are explicit solutions defi ned on the interval (?5, 5). The solution curves given in Figures 1.1.2(b) and 1.1.2(c) are segments of the graph of the implicit solution in Figure 1.1.2(a). Any relation of the form x2? y2 ? c ? 0 formally satisfi es (8) for any constant c. However, it is understood that the relation should always make sense in the real number system; thus, for example, i