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Chapter 1 First-order Differential Equations 1

1.1 Introduction 1

Exercise 1.1 7

1.2 First-order Linear Differential Equations 8

1.2.1 First-order Homogeneous Linear Differential Equations 8

1.2.2 First-order Nonhomogeneous Linear Differential Equations 11

1.2.3 Bernoulli Equations 16

Exercise 1.2 18

1.3 Separable Equations 19

1.3.1 Separable Equations 19

1.3.2 Homogeneous Equations 23

Exercise 1.3 26

1.4 Applications 27

Module 1 The Spread of Technological Innovations 27

Module 2 The Van Meegeren Art Forgeries 30

1.5 Exact Equations 35

1.5.1 Criterion for Exactness 35

1.5.2 Integrating Factor 39

Exercise 1.5 42

1.6 Existence and Uniqueness of Solutions 43

Exercise 1.6 50

Chapter 2 Second-order Differential Equations 51

2.1 General Solutions of Homogeneous Second-order Linear Equations 51

Exercise 2.1 59

2.2 Homogeneous Second-order Linear Equations with Constant Coefficients 60

2.2.1 The Characteristic Equation Has Distinct Real Roots 61

2.2.2 The Characteristic Equation Has Repeated Roots 62

2.2.3 The Characteristic Equation Has Complex Conjugate Roots 63

Exercise 2.2 65

2.3 Nonhomogeneous Second-order Linear Equations 66

2.3.1 Structure of General Solutions 66

2.3.2 Method of Variation of Parameters 68

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

Exercise 2.3 76

2.4 Applications 77

Module 1 An Atomic Waste Disposal Problem 77

Module 2 Mechanical Vibrations 82

Chapter 3 Linear Systems of Differential Equations 90

3.1 Basic Concepts and Theorems 90

Exercise 3.1 98

3.2 The Eigenvalue-Eigenvector Method of Finding Solutions 99

3.2.1 The Characteristic Polynomial of A Has n Distinct Real Eigenvalues 100

3.2.2 The Characteristic Polynomial of A Has Complex Eigenvalues 101

3.2.3 The Characteristic Polynomial of A Has Equal Eigenvalues 104

Exercise 3.2 108

3.3 Fundamental Matrix Solution;Matrix-valued Exponential Function eAt 109

Exercise 3.3 113

3.4 Nonhomogeneous Equations;Variation of Parameters 115

Exercise 3.4 120

3.5 Applications 121

Module 1 The Principle ofCompetitive Exclusion in Population Biology 121

Module 2 A Model for the Blood Glucose Regular System 127

Chapter 4 Laplace Transforms and Their Applications in Solving 136

Differential Equations 136

4.1 Laplace Transforms 136

Exercise 4.1 138

4.2 Properties of Laplace Transforms 138

Exercise 4.2 145

4.3 Inverse Laplace Transforms 146

Exercise 4.3 148

4.4 Solving Differential Equations by Laplace Transforms 148

4.4.1 The Right-Hand Side of the Differential Equation is Discontinuous 152

4.4.2 The Right-Hand Side of Differential Equation is an Impulsive Function 154

Exerci8e 4.4 156

4.5 Solving Systems of Differential Equations by Laplace Transforms 157

Exercise 4.5 159

Chapter 5 Introduction to the Stability Theory 161

5.1 Introduction 161

Exercise 5.1 164

5.2 Stability of the Solutions of Linear System 164

Exercise 5.2 171

5.3 Geometrical Characteristics of Solutions of the System of Differential Equations 173

5.3.1 Phase Space and Direction Field 173

5.3.2 Geometric Characteristics of the Orbits near a Singular Point 176

5.3.3 Stability of Singular Points 180

Exercise 5.3 183

5.4 Applications 183

Module 1 Volterra's Principle 183

Module 2 Mathematical Theories of War 188

Answers to Selected Exercises 196

References 209

附录 软件包Iode简介 210