常微分方程及其应用 理论与模型2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

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Chapter 1 First-order Differential Equations1

1.1 Introduction1

Exercise 1.17

1.2 First-order Linear Differential Equations8

1.2.1 First-order Homogeneous Linear Differential Equations8

1.2.2 First-order Nonhomogeneous Linear Differential Equations11

1.2.3 Bernoulli Equations16

Exercise 1.218

1.3 Separable Equations19

1.3.1 Separable Equations19

1.3.2 Homogeneous Equations23

Exercise 1.326

1.4 Applications27

Module 1 The Spread of Technological Innovations27

Module 2 The Van Meegeren Art Forgeries30

1.5 Exact Equations35

1.5.1 Criterion for Exactness35

1.5.2 Integrating Factor39

Exercise 1.542

1.6 Existence and Uniqueness of Solutions43

Exercise 1.650

Chapter 2 Second-order Differential Equations51

2.1 General Solutions of Homogeneous Second-order Linear Equations51

Exercise 2.159

2.2 Homogeneous Second-order Linear Equations with Constant Coefficients60

2.2.1 The Characteristic Equation Has Distinct Real Roots61

2.2.2 The Characteristic Equation Has Repeated Roots62

2.2.3 The Characteristic Equation Has Complex Conjugate Roots63

Exercise 2.265

2.3 Nonhomogeneous Second-order Linear Equations66

2.3.1 Structure of General Solutions66

2.3.2 Method of Variation of Parameters68

2.3.3 Methods for Some Special Form of the Nonhomogeneous Term g(t)70

Exercise 2.376

2.4 Applications77

Module 1 An Atomic Waste Disposal Problem77

Module 2 Mechanical Vibrations82

Chapter 3 Linear Systems of Differential Equations90

3.1 Basic Concepts and Theorems90

Exercise 3.198

3.2 The Eigenvalue-Eigenvector Method of Finding Solutions99

3.2.1 The Characteristic Polynomial of A Has n Distinct Real Eigenvalues100

3.2.2 The Characteristic Polynomial of A Has Complex Eigenvalues101

3.2.3 The Characteristic Polynomial of A Has Equal Eigenvalues104

Exercise 3.2108

3.3 Fundamental Matrix Solution;Matrix-valued Exponential Function eAt109

Exercise 3.3113

3.4 Nonhomogeneous Equations;Variation of Parameters115

Exercise 3.4120

3.5 Applications121

Module 1 The Principle ofCompetitive Exclusion in Population Biology121

Module 2 A Model for the Blood Glucose Regular System127

Chapter 4 Laplace Transforms and Their Applications in Solving136

Differential Equations136

4.1 Laplace Transforms136

Exercise 4.1138

4.2 Properties of Laplace Transforms138

Exercise 4.2145

4.3 Inverse Laplace Transforms146

Exercise 4.3148

4.4 Solving Differential Equations by Laplace Transforms148

4.4.1 The Right-Hand Side of the Differential Equation is Discontinuous152

4.4.2 The Right-Hand Side of Differential Equation is an Impulsive Function154

Exerci8e 4.4156

4.5 Solving Systems of Differential Equations by Laplace Transforms157

Exercise 4.5159

Chapter 5 Introduction to the Stability Theory161

5.1 Introduction161

Exercise 5.1164

5.2 Stability of the Solutions of Linear System164

Exercise 5.2171

5.3 Geometrical Characteristics of Solutions of the System of Differential Equations173

5.3.1 Phase Space and Direction Field173

5.3.2 Geometric Characteristics of the Orbits near a Singular Point176

5.3.3 Stability of Singular Points180

Exercise 5.3183

5.4 Applications183

Module 1 Volterra's Principle183

Module 2 Mathematical Theories of War188

Answers to Selected Exercises196

References209

附录 软件包Iode简介210