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#### Ordinary Differential Equations and Their Applications - Theories and Models

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Contents
Chapter 1 First-order Dierential Equations 1
1.1 Introduction 1
Exercise 1.1 7
1.2 First-order Linear Dierential Equations 8
1.2.1 First-order Homogeneous Linear Dierential Equations 8
1.2.2 First-order Nonhomogeneous Linear Dierential 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 Dierential 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 Coe±cients 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 Dierential 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 of Competitive 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 Dierential 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 Dierential Equations by Laplace Transforms 148
4.4.1 The Right-Hand Side of the Dierential Equation is Discontinuous 152
4.4.2 The Right-Hand Side of Dierential Equation is an Impulsive Function 154
Exercise 4.4 156
4.5 Solving Systems of Dierential 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 Dierential 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