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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 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.
