微分方程

CHAPTER 1 INTRODUCTION TO DIFFERENTIAL EQUATIONS
1.1 MODELS ON DIFFERENTIAL EQUATIONS
1.2 BASIC CONCEPTS OF DIFFERENTIAL EQUATIONS
1.2.1 Classifications of Differential Equations
1.2.2 Solution of a Differential Equation
1.2.3 Initial-and Boundary-Value Problems
Summary
Exercise
CHAPTER 2 FIRST-ORDER ORDINARY DIFFERENTIAL EQUATIONS
2.1 THREE BASIC TYPES OF FIRST-ORDER EXPLICIT EQUATIONS
2.1.1 Equations in Which the Variables Are Separable
2.1.2 First-Order Linear Differential Equations
2.1.3 Exact Differential Equations
2.2 TWO AVAILABLE TACTICS
2.2.1 Finding Integrating Factors
2.2.2 Use of Substitutions
2.3 FUNDAMENTAL THEORY OF INITIAL-VALUE PROBLEM
2.3.1 Geometric Interpretation of Solutions
2.3.2 Existence and Uniqueness of the Solutions
2.3.3 * Properties of the Solution on Initia-Value
2.4 METHODS OF APPROXIMATION
2.4.1 The Pieard Method
2.4.2 The Cauehy-Euler Method
2.4.3 Taylor Series Method
2.5 FIRST-ORDER IMPLICIT EQUATIONS
2.5.1 Special Methods for First-Order hnplicit Equations
2.5.2 * Singular Solutions and Envelopes
Summary
Exercise
CHAPTER 3 HIGH-ORDER ORDINARY DIFFERENTIAL EQUATIONS
3.1 FUNDAMENTAL THEORIES OF LINEAR EQUATIONS
3.1.1 Preliminary Knowledge for the Linear Equations
3.1.2 Properties of Solutions of the Linear Equations
3.2 LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS
3.2.1 Homogeneous Linear Equations with Constant Coefficients
3.2.2 Solution by Undetermined Coefficients
3.2.3 Solution by Laplace Transform
3.3 LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS
3.3.1 Euler s Equation
3.3.2 Solution by Liouville s Formula
3.3.3 Solution by Variation of Parameters
3.4 SOLUTION BY POWER SERIES
3.4.1 Ordinary and Singular Point of the Equation
3.4.2 Solution at an Ordinary Point of the Equation
3.4.3 * Soh:fion at a Regular Singular Point of the Equation
3.5 OTHER PROBLEMS OF nTH-ORDER EQUATIONS
3.5.1 Linear Boundary-Value Problems
3.5.2 Reducible nth-Order Differential Equations
Summary
Exercise
CHAPTER 4 FIRST-ORDER ORDINARY DIFFERENTIAL SYSTEMS
4.1 FUNDAMENTAL THEORIES OF LINEAR SYSTEMS
4.1.1 Preliminary Knowledge for the Linear Systems
4.1.2 Properties of Solutions of First-Order Linear Systems
4.1.3 Solution Matrix and General Solution Matrix
4.2 HOMOGENEOUS LINEAR SYSTEMS WITH CONSTANT COEFFICIENTS
4.2.1 Solution by Finding a General Solution Matrix
4.2.2 Solution by Finding the Standard Solution Matrix
4.3 NON-HOMOGENEOUS LINEAR SYSTEMS WITH CONSTANT COEFFICIENTS
4.3.1 * Solution by Undetermined Coefficients
4.3.2 * Solution by Variation of Parameters
4.3.3 Solution by Laplace Transform
4.4 THE FIRST INTEGRAL
4.4.1 Basic Concepts and Theories of the First Integral
4.4.2 Solution by Finding the First Integrals
Summary
Exercise
CHAPTER 5 " QUALITATIVE ANALYSIS AND STABILITY OF SOLUTIONS
5.1 INTRODUCTION TO QUALITATIVE ANALYSIS
5.1.1 Differential Dynamic Systems
5.1.2 Equilibrium Point and Closed Trajectory
5.2 STABILITY OF THE TRIVIAL SOLUTION
5.2.1 Concepts of Stability of the Trivial Solution
5.2.2 The Trivial Solution of the Linear System
5.2.3 Method of Linear Approximation
5.2.4 Liapunov s Second Method
5.3 LOCAL ANALYSIS OF TWO-DIMENSIONAL AUTONOMOUS SYSTEMS
5.3.1 Classification of the Equilibrium Points
5.3.2 Closed Trajectory and Limit Cycle
5.4 GLOBAL PHASE DIAGRAM OF TWO-DIMENSIONAL AUTONOMOUS SYSTEMS
5.4.1 Infinite Points of the System
5.4.2 Examples for the Global Phase Diagram
Summary
Exercise
CHAPTER 6 INTRODUCTION TO PAR...

CHAPTER 1 INTRODUCTION TO DIFFERENTIAL EQUATIONS 1.1 MODELS ON DIFFERENTIAL EQUATIONS 1.2 BASIC CONCEPTS OF DIFFERENTIAL EQUATIONS 1.2.1 Classifications of Differential Equations 1.2.2 Solution of a Differential Equation 1.2.3 Initial-and Boundary-Value Problems Summary ExerciseCHAPTER 2 FIRST-ORDER ORDINARY DIFFERENTIAL EQUATIONS 2.1 THREE BASIC TYPES OF FIRST-ORDER EXPLICIT EQUATIONS 2.1.1 Equations in Which the Variables Are Separable 2.1.2 First-Order Linear Dif...