微分方程与边界值问题
2003-1
机械工业出版社
[美] 兹尔（Zill，D.
631
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微分方程问题是工程和应用数学领域的重要问题。本书是作者多年教学经验的总结，示例丰富、内容全面。条理清晰。在编写的过程中，作者一直遵循便于学生理解和记忆的原则，所以本书的内容没有采用过于理论化的方式，而是以直观、易读的方式表述。本书对传统的教学方式和教学内容的各个方面都进行了革新，不仅内容更加吸引读者，同时加强了与现实世界的联系，使传统的教学内容与新知识完美结合。
PREFACE IXACKNOWLEDGMENTS XIIi1 INTRODUCTION TO DIFFERENTIAL EQUATIONS 1 1.1 Definitions and Terminology 2 1.2 Initial-Value Problems 15 1.3 Differential Equations as Mathematical Models 22 Chapter 1 in Review 372 FIRST-ORDER DIFFERENTIAL EQUATIONS 39 2.1 Solution Curves Without the Solution 40 2.2 Separable Variables 51 2.3 Linear Equations 60 2.4 Exact Equations 72 2.5 Solutions by Substitutions 80 2.6 A Numerical Solution 86 Chapter 2 in Review 923 MODELING WITH FIRST-ORDER DIFFERENTIAL EQUATIONS 95 3.1 Linear Equations 96 3.2 Nonlinear Equations 109 3.3 Systems of Linear and Nonlinear Differential Equations 121 Chapter 3 in Review 130 Project Module: Harvesting of Renewable Natural Resources, by Gilbert N. Lewis 1334 HIGHER-ORDER DIFFERENTIAL EQUATIONS 138 4.1 Preliminary Theory: Linear Equations 139 4.1.1 Initial-Value and Boundary-Value Problems 139 4.1.2 Homogeneous Equations 142 4.1.3 Nonhomogeneous Equations 148 4.2 Reduction of Order 154 4.3 Homogeneous Linear Equations with Constant Coefficients 158 4.4 Undetermined Coefficients--Superposition Approach 167 4.5 Undetermined Coefficients--Annihilator Approach 178 4.6 Variation of Parameters 188 4.7 Cauchy-Euler Equation 193 4.8 Solving Systems of Linear Equations by Elimination 201 4.9 Nonlinear Equations 207 Chapter 4 in Review 2125 MODELING WITH HIGHER-ORDER DIFFERENTIAL EQUATIONS 215 5.1 Linear Equations: Initial-Value Problems 216 5.1.1 Spring/Mass Systems: Free Undamped Motion 216 5.1.2 Spring/Mass Systems: Free Damped Motion 220 5.1.3 Spring/Mass Systems: Driven Motion 224 5.1.4 Series Circuit Analogue 227 5.2 Linear Equations: Boundary-Value Problems 237 5.3 Nonlinear Equations 247 Chapter 5 in Review 259 Project Module: The Collapse of the Tacoma Narrows Suspension Bridge, by Gilbert N. Lewis 2636 SERIES SOLUTIONS Of LINEAR EQUATIONS 267 6.1 Solutions About Ordinary Points 268 6.1.1 Review of Power Series 268 6.1.2 Power Series Solutions 271 6.2 Solutions About Singular Points 280 6.3 Two Special Equations 292 Chapter 6 in Review 3047 THE LAPLACE TRANSFORM 306 7.1 Definition of the Laplace Transform 307 7.2 Inverse Transform and Transforms of Derivatives 314 7.3 Translation Theorems 324 7.3.1 Translation on the s-Axis 324 7.3.2 Translation on the t-Axis 328 7.4 Additional Operational Properties 338 7.5 Dirac Delta Function 351 7.6 Systems of Linear Equations 354 Chapter 7 in Review 3618 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS 364 8.1 Preliminary Theory 365 8.2 Homogeneous Linear Systems with Constant Coefficients 375 8.2.1 Distinct Real Eigenvalues 376 8.2.2 Repeated Eigenvalues 380 8.2.3 Complex Eigenvalues 384 8.3 Variation of Parameters 393 8.4 Matrix Exponential 399 Chapter 8 in Review 404 Project Module: Earthquake Shaking of Multistory Buildings, by Gilbert N. Lewis 4069 NUMERICAL SOLUTIONS Of ORDINARY DIFFERENTIAL EQUATIONS 410 9.1 Euler Methods and Error Analysis 411 9.2 Runge-Kutta Methods 417 9.3 Multistep Methods 424 9.4 Higher-Order Equations and Systems 427 9.5 Second-Order Boundary-Value Problems 433 Capter 9 in Review 43810 PLANEAUTONOMOUSSYSTEMSAND STABILITY 439 10.1 Autonomous Systems, Critical Points, and Periodic Solutions 440 10.2 Stability of Linear Systems 448 10.3 Linearization and Local Stability 458 10.4 Modeling Using Autonomous Systems 470 Chapter 10 in Review 48011 ORTHOGONAL FUNCTIONSAND FOURIER SERIES 483 11.1 Orthogonal Functions 484 11.2 Fourier Series 489 11.3 Fourier Cosine and Sine Series 495 11.4 Sturm-Liouville Problem 504 11.5 Bessel and Legendre Series 511 11.5.1 Fourier-Bessel Series 512 11.5.2 Fourier-Legendre Series 515 Chapter 11 in Review 519 PARTIAL DIFFERENTIAL EQUATIONS AND12 BOUNDARY-VALUEPROBLEMS IN RECTANGULAR COORDINATES 521 12.1 Separable Partial Differential Equations 522 12.2 Classical Equations and Boundary-Value Problems 527 12.3 Heat Equation 533 12.4 Wave Equation 536 12.5 Laplace's Equation 542 12.6 Nonhomogeneous Equations and Boundary Conditions 547 12.7 Orthogonal Series Expansions 551 12.8 Boundary-Value Problems Involving Fourier Series in Two Variables 555 Chapter 12 in Review 55913 BOUNDARY-VALUE PROBLEMS IN OTHER COORDINATE SYSTEMS 561 13.1 Problems Involving Laplace's Equation in Polar Coordinates 562 13.2 Problems in Polar and Cylindrical Coordinates: Bessel Functions 567 13.3 Problems in Spherical Coordinates: Legendre Polynomials 575 Chapter 13 in Review 57814 INTEGRAL TRANSFORM METHOD 581 14.1 Error Function 582 14.2 Applications of the Laplace Transform 584 14.3 Fourier Integral 595 14.4 Fourier Transforms 601 Chapter 14 in Review 60715 NUMERICAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS 610 15.1 Elliptic Equations 611 15.2 Parabolic Equations 617 15.3 Hyperbolic Equations 625Chapter 15 in Review 630APPENDIXES APP-1I Gamma Function APP-1II Introduction to Matrices APP-3III Laplace Transforms APP-25SELECTED ANSWERS FOR ODD-NUMBEREDPROBLEMS AN- 1INDEX I-1
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