1、Licensed to:Differential Equations withBoundary-Value Problems,Seventh EditionDennis G. Zill and Michael R. CullenExecutive Editor: Charlie Van WagnerDevelopment Editor: Leslie LahrAssistant Editor: Stacy GreenEditorial Assistant: Cynthia AshtonTechnology Project Manager: Sam SubityMarketing Special
2、ist: Ashley PickeringMarketing Communications Manager:Darlene Amidon-BrentProject Manager, Editorial Production: Cheryll LinthicumCreative Director: Rob HugelArt Director: Vernon BoesPrint Buyer: Rebecca CrossPermissions Editor: Mardell Glinski SchultzProduction Service: Hearthside Publishing Servic
3、esText Designer: Diane BeasleyPhoto Researcher: Don SchlotmanCopy Editor: Barbara WilletteIllustrator: Jade Myers, MatrixCover Designer: Larry DidonaCover Image: Getty ImagesCompositor: ICC Macmillan Inc. 2009, 2005 Brooks/Cole, Cengage LearningALL RIGHTS RESERVED. No part of this work covered by th
4、ecopyright herein may be reproduced, transmitted, stored, or usedin any form or by any means graphic, electronic, or mechanical,including but not limited to photocopying, recording, scanning,digitizing, taping, Web distribution, information networks, orinformation storage and retrieval systems, exce
5、pt 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: 2008924835ISBN-13: 978-0-495-10836-8ISBN-10: 0-495-10836-7Brooks/Cole10 Davis DriveBelmont, CA 94002-3098USACengage Learning is
6、a leading provider of customized learningsolutions with office locations around the globe, including Singapore,the United Kingdom, Australia, Mexico, Brazil, and Japan. Locateyour local office at Learning products are represented in Canada by Nelson Education, Ltd.For your course and learning solut
7、ions, 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 atCengage Learning Customer the fourth derivative iswritten y(4)instead of yH11033H11033. In general, the nth derivative of y is writt
8、en dnyH20862dxnor 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 andindependent variables. For example, in the equationit is immediately seen that the symbol x now represents a depen
9、dent variable,whereas the independent variable is t. You should also be aware that in physicalsciences and engineering, Newtons dot notation (derogatively referred to by someas the “flyspeck” notation) is sometimes used to denote derivatives with respectto time t. Thus the differential equation d2sH
10、20862dt2H11005H1100232 becomes s H11005H1100232. Partialderivatives are often denoted by a subscript notation indicating the indepen-dent variables. For example, with the subscript notation the second equation in(3) becomes uxxH11005 uttH11002 2ut.CLASSIFICATION BY ORDER The order of a differential
11、equation (eitherODE or PDE) is the order of the highest derivative in the equation. For example,is a second-order ordinary differential equation. First-order ordinary differentialequations are occasionally written in differential form M(x, y) dx H11001 N(x, y) dy H11005 0.For example, if we assume t
12、hat y denotes the dependent variable in (y H11002 x) dx H11001 4xdyH11005 0, then yH11032H11005dyH20862dx, so by dividing by the differential dx, weget the alternative form 4xyH11032H11001y H11005 x. See the Remarks at the end of this section.In symbols we can express an nth-order ordinary different
13、ial equation in onedependent variable by the general form, (4)where F is a real-valued function of n H11001 2 variables: x, y, yH11032,., y(n). For both prac-tical and theoretical reasons we shall also make the assumption hereafter that it ispossible to solve an ordinary differential equation in the
14、 form (4) uniquely for theF(x, y, yH11032, . . . , y(n) H11005 0first ordersecond orderH11001 5()3H11002 4y H11005 exdydxd2ydx2d2xdt2H11001 16x H11005 0unknown functionor dependent variableindependent variableH111282uH11128x2H11001H111282uH11128y2H11005 0, H111282uH11128x2H11005H111282uH11128t2H1100
15、2 2H11128uH11128t, and H11128uH11128yH11005H11002H11128vH11128x1.1 DEFINITIONS AND TERMINOLOGY3*Except for this introductory section, only ordinary differential equations are considered in A FirstCourse in Differential Equations with Modeling Applications, Ninth Edition. In that text theword equatio
16、n and the abbreviation DE refer only to ODEs. Partial differential equations or PDEsare 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 3Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May
17、 not be copied, scanned, or duplicated, in whole or in part.Licensed to:highest derivative y(n)in terms of the remaining n H11001 1 variables. The differentialequation, (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
18、 use the normal formsto represent general first- and second-order ordinary differential equations. For example,the normal form of the first-order equation 4xyH11032H11001y H11005 x is yH11032H11005(x H11002 y)H208624x; the normalform of the second-order equation yH11033H11002yH11032H110016y H11005 0
19、 is yH11033H11005yH11032H110026y. See the Remarks.CLASSIFICATION BY LINEARITY An nth-order ordinary differential equation (4)is said to be linear if F is linear in y, yH11032,.,y(n). This means that an nth-order ODE islinear when (4) is an(x)y(n)H11001 anH110021(x)y(nH110021)H11001H11080H11080H11080
20、H11001a1(x)yH11032H11001a0(x)y H11002 g(x) H11005 0 or. (6)Two important special cases of (6) are linear first-order (n H11005 1) and linear second-order (n H11005 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
21、 ODE are as follows: The dependent variable y and all its derivatives yH11032, yH11033, ., y(n)are of thefirst degree, that is, the power of each term involving y is 1. The coefficients a0, a1, ., anof y, yH11032, ., y(n)depend at most on theindependent variable x.The equationsare, in turn, linear f
22、irst-, second-, and third-order ordinary differential equations. Wehave just demonstrated that the first equation is linear in the variable y by writing it inthe alternative form 4xyH11032H11001y H11005 x. A nonlinear ordinary differential equation is sim-ply one that is not linear. Nonlinear functi
23、ons of the dependent variable or its deriva-tives, such as sin y or , cannot appear in a linear equation. Thereforeare examples of nonlinear first-, second-, and fourth-order ordinary differential equa-tions, respectively.SOLUTIONS As was stated before, one of the goals in this course is to solve, o
24、rfind solutions of, differential equations. In the next definition we consider the con-cept of a solution of an ordinary differential equation.nonlinear term:coefficient depends on ynonlinear term:nonlinear function of ynonlinear term:power not 1(1 H11002 y)yH11032 H11001 2y H11005 ex, H11001 sin y
25、H11005 0, andd2ydx2H11001 y2H11005 0d4ydx4eyH11032(y H11002 x)dx H11001 4x dy H11005 0, yH11033H110022yH11032H11001y H11005 0, and d3ydx3H11001 x dydxH11002 5y H11005 exa1(x) dydxH11001 a0(x)y H11005 g(x) and a2(x) d2ydx2H11001 a1(x) dydxH11001 a0(x)y H11005 g(x)an(x) dnydxnH11001 anH110021(x) dnH11
26、0021ydxnH110021H11001H11080 H11080 H11080H11001a1(x) dydxH11001 a0(x)y H11005 g(x)dydxH11005 f (x, y) and d2ydx2H11005 f (x, y, yH11032)dnydxnH11005 f (x, y, yH11032, . . . , y(nH110021)4CHAPTER 1 INTRODUCTION TO DIFFERENTIAL EQUATIONS08367_01_ch01_p001-033.qxd 4/7/08 1:04 PM Page 4Copyright 2009 Ce
27、ngage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.Licensed to:DEFINITION 1.1.2 Solution of an ODEAny function H9278, defined on an interval I and possessing at least n derivativesthat are continuous on I, which when substituted into an nth-order
28、 ordinarydifferential equation reduces the equation to an identity, is said to be asolution of the equation on the interval.In other words, a solution of an nth-order ordinary differential equation (4) is a func-tion H9278 that possesses at least n derivatives and for whichWe say that H9278 satisfie
29、s the differential equation on I. For our purposes we shall alsoassume that a solution H9278 is a real-valued function. In our introductory discussion wesaw that is a solution of dyH20862dx H11005 0.2xy on the interval (H11002H11009, H11009).Occasionally, it will be convenient to denote a solution b
30、y the alternativesymbol y(x).INTERVAL OF DEFINITION You cannot think solution of an ordinary differentialequation without simultaneously thinking interval. The interval I in Definition 1.1.2is variously called the interval of definition, the interval of existence, the intervalof validity, or the dom
31、ain of the solution and can be an open interval (a, b), a closedinterval a, b, an infinite interval (a, H11009), and so on.EXAMPLE 1 Verification of a SolutionVerify that the indicated function is a solution of the given differential equation onthe interval (H11002H11009, H11009).(a) (b)SOLUTION One
32、 way of verifying that the given function is a solution is to see, aftersubstituting, whether each side of the equation is the same for every x in the interval.(a) Fromwe see that each side of the equation is the same for every real number x. Notethat is, by definition, the nonnegative square root o
33、f .(b) From the derivatives yH11032H11005xexH11001 exand yH11033H11005xexH11001 2exwe have, for every realnumber x,Note, too, that in Example 1 each differential equation possesses the constant so-lution y H11005 0, H11002H11009 H11021 x H11021H11009. A solution of a differential equation that is id
34、enticallyzero on an interval I is said to be a trivial solution.SOLUTION CURVE The graph of a solution H9278 of an ODE is called a solutioncurve. Since H9278 is a differentiable function, it is continuous on its interval I of defini-tion. Thus there may be a difference between the graph of the funct
35、ion H9278 and theright-hand side: 0.left-hand side: yH11033H110022yH11032H11001y H11005 (xexH11001 2ex) H11002 2(xexH11001 ex) H11001 xexH11005 0,116 x4y1/2H1100514 x2right-hand side: xy1/2H11005 x H11554H20898116x4H208991/2H11005 x H11554H2089814x2H20899H1100514x3,left-hand side: dydxH11005116(4 H1
36、1554 x3) H1100514x3,yH11033H110022yH11032H11001y H11005 0; y H11005 xexdydx H11005 xy1/2; y H11005116 x4y H11005 e0.1x2F(x, H9278(x), H9278H11032(x), . . . , H9278(n)(x) H11005 0 for all x in I.1.1 DEFINITIONS AND TERMINOLOGY508367_01_ch01_p001-033.qxd 4/7/08 1:04 PM Page 5Copyright 2009 Cengage Lea
37、rning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.Licensed to:graph of the solution H9278. Put another way, the domain of the function H9278 need not bethe same as the interval I of definition (or domain) of the solution H9278. Example 2illustrates the d
38、ifference.EXAMPLE 2 Function versus SolutionThe domain of y H11005 1H20862x, considered simply as a function, is the set of all real num-bers x except 0. When we graph y H11005 1H20862x, we plot points in the xy-plane corre-sponding to a judicious sampling of numbers taken from its domain. The ratio
39、nalfunction y H11005 1H20862x 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 H11005 1H20862x is not differentiable at x H11005 0,since the y-axis (whose equation is x H11005 0) is a vertical asymptote of the graph.Now y H11005 1H20
40、862x is also a solution of the linear first-order differential equationxyH11032H11001y H11005 0. (Verify.) But when we say that y H11005 1H20862x is a solution of this DE, wemean that it is a function defined on an interval I on which it is differentiable andsatisfies the equation. In other words, y
41、 H11005 1H20862x is a solution of the DE on any inter-val that does not contain 0, such as (H110023, H110021), , (H11002H11009, 0), or (0, H11009). Becausethe solution curves defined by y H11005 1H20862x for H110023 H11021 x H11021H110021 and are sim-ply segments, or pieces, of the solution curves d
42、efined by y H11005 1H20862x for H11002H11009 H11021 x H11021 0and 0 H11021 x H11021H11009, respectively, it makes sense to take the interval I to be as large aspossible. Thus we take I to be either (H11002H11009, 0) or (0, H11009). The solution curve on (0, H11009)is shown in Figure 1.1.1(b).EXPLICI
43、T AND IMPLICIT SOLUTIONS You should be familiar with the termsexplicit functions and implicit functions from your study of calculus. A solution inwhich the dependent variable is expressed solely in terms of the independentvariable and constants is said to be an explicit solution. For our purposes, l
44、et usthink of an explicit solution as an explicit formula y H11005 H9278(x) that we can manipulate,evaluate, and differentiate using the standard rules. We have just seen in the last twoexamples that , y H11005 xex, and y H11005 1H20862x are, in turn, explicit solutionsof dyH20862dx H11005 xy1/2, yH
45、11033H110022yH11032H11001y H11005 0, and xyH11032H11001y H11005 0. Moreover, the trivial solu-tion y H11005 0 is an explicit solution of all three equations. When we get down tothe business of actually solving some ordinary differential equations, you willsee that methods of solution do not always l
46、ead directly to an explicit solutiony H11005 H9278(x). This is particularly true when we attempt to solve nonlinear first-orderdifferential equations. Often we have to be content with a relation or expressionG(x, y) H11005 0 that defines a solution H9278 implicitly.DEFINITION 1.1.3 Implicit Solution
47、 of an ODEA relation G(x, y) H11005 0 is said to be an implicit solution of an ordinarydifferential equation (4) on an interval I, provided that there exists at leastone function H9278 that satisfies the relation as well as the differential equationon I.It is beyond the scope of this course to inves
48、tigate the conditions under which arelation G(x, y) H11005 0 defines a differentiable function H9278. So we shall assume that ifthe formal implementation of a method of solution leads to a relation G(x, y) H11005 0,then there exists at least one function H9278 that satisfies both the relation (that
49、is,G(x, H9278(x) H11005 0) and the differential equation on an interval I. If the implicit solutionG(x, y) H11005 0 is fairly simple, we may be able to solve for y in terms of x and obtainone or more explicit solutions. See the Remarks.y H11005116 x412H11021 x H11021 10(12, 10)6CHAPTER 1 INTRODUCTION TO DIFFERENTIAL EQUATIONS1xy1(a) function y H11005 1/x, x HS33527 0(b) solution y H11005 1/x, (0, L50525)1xy1FIGURE 1.1.1 The function y H11005 1H20862xis not the same a
