华章数学译丛 微分方程与边界值问题 英文.pdf
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1、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
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3、ice: 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 RES
4、ERVED. 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 i
5、nformation 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
6、 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
7、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
8、)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 i
9、s 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 derivat
10、ives 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
11、 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.
12、 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 equatio
13、n 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) u
14、niquely 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 i
15、ntroductory 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 expa
16、nded 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
17、 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 eq
18、uations. 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 l
19、inear 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
20、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?
21、, . . . , 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 nonli
22、near 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,
23、 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: no
24、nlinear 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
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