微分方程及邊值問題

^[1]  本書以一些模型問題為背景，藉助於數學軟件Maple，Mathematica 及MATLAB，利用符號運算、圖像表示和數值解法等手段，系統地介紹了（線性與非線性）微分方程的基本概念和基本方法。通過40多個實際模型的討論，使讀者對建模、求解、分析解所反映的性質這一過程進行全面的瞭解。利用Maple，Mathematica及MATLAB在圖形顯示、符號計算、數值計算方面的功能，定性地分析了微分方程解的性質，700餘幅圖將方向場、解曲線、相平面等概念形象直觀地表示出來。另外，書中選配了1900餘道習題供讀者使用。 本書可供學習數學建模或微分方程的學生作為參考書，對於從事計算與建模的科技人員，本書也具有很高的價值。 ^[1] 

Application Modules

Preface

CHAPTER 1 First-Order Differential Equations

1.1 Differential Equations and Mathematical Models

1.2 Integrals as General and Paticular Solutions

1.3 Slope Fields and Solution Curves

1.4 Separable Equations and Applications

1.5 Linear First-Order Equations

1.6 Substifution Methods and Exact Equations

CHAPTER 2 Mathematical Models and Numercal Methods

2.1 Population Models

2.2 Equlilibrium Solutions and Stablility

2.3 Acceleration-Velocity Models

2.4 Numerical Approzimation:Euler's Method

2.5 A Closer Look at the Euler Method

2.6 The Runge-Kutta Method

CHAPTER 3 Linear Equations of Higher Order

3.1 Introduction:Second-Order Linear Equations

3.2 General Solutions of Linear Equations

3.3 Homogeneous Equations with Constant Coelficients

3.4 Mechanical Vibrations

3.5 Nonhomogeneous Equations and Undetermined Coefficients

3.6 Forced Oscillations and Resonance

3.7 Electrical Ciruits

3.8 Endpoint Problems and Eigenvalues

CHAPTER 4 Introduction to Systems of Differential Equations

4.1 First-Order Systems and Applications

4.2 The Method of Elimination

4.3 Numerical Methods for Systems

CHAPTER 5 Linear Systems of Differential Equtions

5.1 Matrices and Linear Systems

5.2 The Eigenvalue Method for Homogeneous Systems

5.3 Second-Order Systems and Mechanical Applications

5.4 Multiple Eigenvaluve Solutions

5.5 Matrix Exponentials and Linear Systems

5.6 Nonhomogeneous Linear Systems

CHAPTER 6 Nonlinear Systems and Phenomena

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CHATPER 7 Laplace Transform Methods

CHAPTER 8 Power Series Methods

CHAPTER 9 Fourier Series Methods

CHATPER 10 Eigenvalues and Boundary Value Problems

References for Futher Study

Appendix:Existence and Uniqueness of Solutions

Answers to Selected Problems

Index ^[1]