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3、ifferentialEquationsDemystifiedSTEVEN G. KRANTZMcGRAW-HILLNew York Chicago San Francisco Lisbon LondonMadrid Mexico City Milan New Delhi San JuanSeoul Singapore Sydney TorontoCopyright 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Manufactured in the United States ofAmerica. Except as
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14、ct, tort or otherwise. DOI: 10.1036/0071440259We hope you enjoy thisMcGraw-Hill eBook! Ifyoud like more information about this book,its author, or related books and websites,please click here.ProfessionalWant to learn more?CONTENTSPreface ixCHAPTER 1 What Is a Differential Equation? 11.1 Introductor
15、y Remarks 11.2 The Nature of Solutions 41.3 Separable Equations 71.4 First-Order Linear Equations 101.5 Exact Equations 131.6 Orthogonal Trajectories and Familiesof Curves 191.7 Homogeneous Equations 221.8 Integrating Factors 261.9 Reduction of Order 301.10 The Hanging Chain and Pursuit Curves 361.1
16、1 Electrical Circuits 43Exercises 46CHAPTER 2 Second-Order Equations 482.1 Second-Order Linear Equations withConstant Coefficients 482.2 The Method of UndeterminedCoefficients 542.3 The Method of Variation of Parameters 58vFor more information about this title, click hereCONTENTSvi2.4 The Use of a K
17、nown Solution toFind Another 622.5 Vibrations and Oscillations 652.6 Newtons Law of Gravitation andKeplers Laws 752.7 Higher-Order Linear Equations,Coupled Harmonic Oscillators 85Exercises 90CHAPTER 3 Power Series Solutions andSpecial Functions 923.1 Introduction and Review ofPower Series 923.2 Seri
18、es Solutions of First-OrderDifferential Equations 1023.3 Second-Order Linear Equations:Ordinary Points 106Exercises 113CHAPTER 4 Fourier Series: Basic Concepts 1154.1 Fourier Coefficients 1154.2 Some Remarks About Convergence 1244.3 Even and Odd Functions: Cosine andSine Series 1284.4 Fourier Series
19、 on Arbitrary Intervals 1324.5 Orthogonal Functions 136Exercises 139CHAPTER 5 Partial Differential Equations andBoundary Value Problems 1415.1 Introduction and Historical Remarks 1415.2 Eigenvalues, Eigenfunctions, andthe Vibrating String 1445.3 The Heat Equation: FouriersPoint of View 1515.4 The Di
20、richlet Problem for a Disc 1565.5 SturmLiouville Problems 162Exercises 166CONTENTS viiCHAPTER 6 Laplace Transforms 1686.1 Introduction 1686.2 Applications to DifferentialEquations 1716.3 Derivatives and Integrals ofLaplace Transforms 1756.4 Convolutions 1806.5 The Unit Step and ImpulseFunctions 189E
21、xercises 196CHAPTER 7 Numerical Methods 1987.1 Introductory Remarks 1997.2 The Method of Euler 2007.3 The Error Term 2037.4 An Improved Euler Method 2077.5 The RungeKutta Method 210Exercises 214CHAPTER 8 Systems of First-Order Equations 2168.1 Introductory Remarks 2168.2 Linear Systems 2198.3 Homoge
22、neous Linear Systems withConstant Coefficients 2258.4 Nonlinear Systems: VolterrasPredatorPrey Equations 233Exercises 238Final Exam 241Solutions to Exercises 271Bibliography 317Index 319This page intentionally left blank PREFACEIf calculus is the heart of modern science, then differential equations
23、are its guts.All physical laws, from the motion of a vibrating string to the orbits of the plan-ets to Einsteins field equations, are expressed in terms of differential equations.Classically, ordinary differential equations described one-dimensional phenom-ena and partial differential equations desc
24、ribed higher-dimensional phenomena.But, with the modern advent of dynamical systems theory, ordinary differentialequations are now playing a role in the scientific analysis of phenomena in alldimensions.Virtually every sophomore science student will take a course in introductoryordinary differential
25、 equations. Such a course is often fleshed out with a brieflook at the Laplace transform, Fourier series, and boundary value problems forthe Laplacian. Thus the student gets to see a little advanced material, and somehigher-dimensional ideas, as well.As indicated in the first paragraph, differential
26、 equations is a lovely venuefor mathematical modeling and the applications of mathematical thinking. Trulymeaningful and profound ideas from physics, engineering, aeronautics, statics,mechanics, and other parts of physical science are beautifully illustrated withdifferential equations.We propose to
27、write a text on ordinary differential equations that will be mean-ingful, accessible, and engaging for a student with a basic grounding in calculus(for example, the student who has studied Calculus Demystified by this authorwill be more than ready for Differential Equations Demystified). There will
28、bemany applications, many graphics, a plethora of worked examples, and hun-dreds of stimulating exercises. The student who completes this book will beixCopyright 2005 by The McGraw-Hill Companies, Inc. Click here for terms of use.PREFACExready to go on to advanced analytical work in applied mathemat
29、ics, engineer-ing, and other fields of mathematical science. It will be a powerful and usefullearning tool.Steven G. Krantz1CHAPTERWhat Is aDifferentialEquation?1.1 Introductory RemarksA differential equation is an equation relating some function f to one or more ofits derivatives. An example isd2fd
30、x2+ 2xdfdx+ f2(x) = sin x. (1)Observe that this particular equation involves a function f together with its firstand second derivatives. The objective in solving an equation like (1) is to find the1Copyright 2005 by The McGraw-Hill Companies, Inc. Click here for terms of use.CHAPTER 1 Differential E
31、quations2function f . Thus we already perceive a fundamental new paradigm: When wesolve 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 naturein physics, in engineering, in chemistr
32、y, in biology,and in astronomyfind their most natural expression in the language of differentialequations. Put in other words, differential equations are the language of nature.Applications of differential equations also abound in mathematics itself, especiallyin geometry and harmonic analysis and m
33、odeling. Differential equations occur ineconomics and systems science and other fields of mathematical science.It is not difficult to perceive why differential equations arise so readily in thesciences. If y = f(x)is a given function, then the derivative df/dx can be inter-preted as the rate of chan
34、ge of f with respect to x. In any process of nature, thevariables involved are related to their rates of change by the basic scientific princi-ples that govern the processthat is, by the laws of nature. When this relationshipis expressed in mathematical notation, the result is usually a differential
35、 equation.Certainly Newtons Law of Universal Gravitation, Maxwells field equations, themotions of the planets, and the refraction of light are important physical exampleswhich can be expressed using differential equations. Much of our understandingof nature comes from our ability to solve differenti
36、al equations. The purpose of thisbook is to introduce you to some of these techniques.The following example will illustrate some of these ideas.According to Newtonssecond law of motion, the acceleration a of a body of mass m is proportional tothe total force F acting on the body. The standard implem
37、entation of this relation-ship isF = m a. (2)Suppose in particular that we are analyzing a falling body of mass m. Expressthe height of the body from the surface of the Earth as y(t) feet at time t. Theonly force acting on the body is that due to gravity. If g is the acceleration dueto gravity (abou
38、t 32 ft/sec2near the surface of the Earth) then the force exertedon the body is m g. And of course the acceleration is d2y/dt2. Thus Newtonslaw (2) becomesm g = m d2ydt2(3)org =d2ydt2.We may make the problem a little more interesting by supposing that air exertsa resisting force proportional to the
39、velocity. If the constant of proportionality is k,CHAPTER 1 Differential Equations 3then the total force acting on the body is mg k (dy/dt). Then the equation (3)becomesm g k dydt= m d2ydt2. (4)Equations (3) and (4) express the essential attributes of this physical system.A few additional examples o
40、f differential equations are these:(1 x2)d2ydx2 2xdydx+ p(p + 1)y = 0; (5)x2d2ydx2+ xdydx+ (x2 p2)y = 0; (6)d2ydx2+ xy = 0; (7)(1 x2)yprimeprime xyprime+ p2y = 0; (8)yprimeprime 2xyprime+ 2py = 0; (9)dydx= k y. (10)Equations (5)(9) are called Legendres equation, Bessels equation, Airysequation, Cheb
41、yshevs equation, and Hermites equation respectively. Each hasa vast literature and a history reaching back hundreds of years. We shall touchon each of these equations later in the book. Equation (10) is the equation ofexponential decay (or of biological growth).Math Note: A great many of the laws of
42、 nature are expressed as second-order differential equations. This fact is closely linked to Newtons second law,which expresses force as mass time acceleration (and acceleration is a secondderivative). But some physical laws are given by higher-order equations. TheEulerBernoulli beam equation is fou
43、rth-order.Each of equations (5)(9) is of second-order, meaning that the highest deriva-tive that appears is the second. Equation (10) is of first-order, meaning that thehighest derivative that appears is the first. Each equation is an ordinary differen-tial equation, meaning that it involves a funct
44、ion of a single variable and theordinary derivatives (not partial derivatives) of that function.CHAPTER 1 Differential Equations41.2 The Nature of SolutionsAn ordinary differential equation of order n is an equation involving an unknownfunction f together with its derivativesdfdx,d2fdx2,.,dnfdxn.We
45、might, in a more formal manner, express such an equation asFparenleftbiggx,y,dfdx,d2fdx2,.,dnfdxnparenrightbigg= 0.Howdoweverifythatagivenfunctionf isactuallythesolutionofsuchanequation?The answer to this question is best understood in the context of concreteexamples.e.g. EXAMPLE 1.1Consider the dif
46、ferential equationyprimeprime 5yprime+ 6y = 0.Without saying how the solutions are actually found, we can at least check thaty1(x) = e2xand y2(x) = e3xare both solutions.To verify this assertion, we note thatyprimeprime1 5yprime1+ 6y1= 2 2 e2x 5 2 e2x+ 6 e2x=4 10 + 6e2x 0andyprimeprime2 5yprime2+ 6y
47、2= 3 3 e3x 5 3 e3x+ 6 e3x=9 15 + 6e3x 0.This process, of verifying that a function is a solution of the given differentialequation, is most likely entirely new for you. You will want to practice and becomeaccustomed to it. In the last example, you may check that any function of the formy(x) = c1e2x+
48、 c2e3x(1)(where c1, c2are arbitrary constants) is also a solution of the differential equation.CHAPTER 1 Differential Equations 5Math Note: This last observation is an instance of the principle of superpositionin physics. Mathematicians refer to the algebraic operation in equation (1) as“taking a li
49、near combination of solutions” while physicists think of the processas superimposing forces.An important obverse consideration is this: When you are going through theprocedure 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 havebeen eliminated from the equation. For then you will have y expressed in terms ofx (at least implicitly). Thus you will have found the sought-after function.For a large
