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 Answers to Selected Exercises 196 References 209 附录 软件包Iode简介 210
京公网安备 11010102004214号