Ordinary Differential Equations

Overview

Ordinary differential equations is a required core course for mathematics and applied mathematics majors, featuring strong theoretical and practical aspects. Bilingual teaching not only enhances students' ability to learn professional knowledge in English but also lays a foundation for further studies by developing their professional skills. Through this course, students are expected to master the basic concepts, theories, and methods of ordinary differential equations, while also cultivating their computational skills. By analyzing typical cases where differential equations have successfully solved real-world problems, students will develop the ability to use differential equations to establish mathematical models and apply Matlab software to address practical issues. This course provides an important platform for fostering students' innovative capabilities and develops their ability to analyze and solve real-world problems using the knowledge they have acquired. It helps students realize that mathematics originates from practice and serves practical purposes, facilitating their transition from "learning mathematics" to "applying mathematics."

Syllabus

Chapter 1

1.1 Introduction
1.2 Basic Concepts about Differential Equations
1.3 Isoclines
1.4 Equations and Differentials

Chapter 2

2.1 Separation of Variables
2.2 Equations with Homogeneous Coefficients
2.3 Equations Reducible to the Homogeneous Coefficients Cases
2.4 Exact Differential Equations
2.5 Equations Reducible to the Exact Equation
2.6 Linear Equations of First Order-Method of Integral Factors
2.7 Linear equations of first order—method of variation of parameters
2.8 Theory of Superposition
2.9 Equations Reducible to the Linear Case: Bernoulli Equation
2.10 Linearly Separable Equations

Chapter 3

3.1 Direction Fields
3.2 Existence and Uniqueness for First Order Equations
3.3 First Order Autonomous Equations-the Qualitative Approach
3.4 First Order Autonomous Equations-Stability of Equilibria
3.5 First Order Autonomous Equations-Bifurcations of Equilibria
3.6 Modeling in Population Biology
3.7 Numerical approximation

Chapter 4

4.1 Introduction of linear differential Equations
4.2 Linearly Dependent and Independent Functions
4.3 Second Order Linear Equation: General Theory
4.4 Homogeneous Linear Equations of Second Order with Constant Coefficients
4.5 Nonhomogeneous Linear Differential Equations of Second Order: Method of Variation of Parameters
4.6 Method of Undetermined Coefficients

Chapter 5

5.1 Introduction to Higher-Order Equations
5.2 Linear Independence and the Wronskian
5.3 Homogeneous Equations with Constant Coefficients
5.4 Homogeneous Equations with Constant Coefficients II
5.5 Nonhomogeneous Equations with Constant Coefficients: Method of Undetermined Coefficients via Superposition
5.6 Nonhomogeneous Equations with Constant Coefficients: Method of Undetermined Coefficients via Annihilation
5.7 Exponential Response and Complex Replacement I
5.8 Exponential Response and Complex Replacement II
5.9 Exponential Response and Complex Replacement III
5.10 Euler-Cauchy Equations I
5.11 Euler-Cauchy Equations II
5.12 Reduction of Order of Linear Differential Equations

Chapter 6

6.1 Useful Terminology
6.2 Gaussian Elimination
6.3 Vector Spaces and Subspaces
6.4 The Null Space and Column Space
6.5 Eigenvalues and Eigenvectors I
6.6 Eigenvalues and Eigenvectors II
6.7 Solving Systems with Real and Distinct or Complex Eigenvalues
6.8 Solving Systems with Repeated Real Eigenvalues
6.9 Matrix Exponentials
6.10 Solving Linear Nonhomogeneous Systems of Equations-Constant Coefficients and Constant Forcing