微分方程

Abstract

課程首頁

本課程是由交通大學應用數學系提供。

Mathematical models as differential Equations – qualitative and quantative analysis.

First order differential equations.

Second order linear differential equations.

Higher order linear differential equations.

Second order nonlinear differential equations – pendulum motion.

課程目標/概述

Mathematical models as differential Equations – qualitative and quantative analysis.

First order differential equations.

Second order linear differential equations.

Higher order linear differential equations.

Second order nonlinear differential equations – pendulum motion.

課程章節

老師授課主題

課本參考章節

Introduction of ODE

Introduction of ODE

Big Picture for ODE

1.8 Linear Equations

1st order linear system

1.8 Linear Equations 1.9 Integrating Factors for Linear Equations

1st order linear system: 量的分析

3.6 Second-Order Linear Equations 4.1 Forced Harmonic Oscillators 4.2 Sinusoidal Forcing

2nd order linear equation with constant coefficeints

3.6 Second-Order Linear Equations

Practical Problem

1.1 Modeling via Differential Equations

Two more “derivation” of math models from:

Predator-Prey problem

Deposit and Withdraw problem

2.1 Modeling via Systems

1.2 ODE Big Picture

1.2 Analytic Technique: Separation of Variables

1.2 Mixed Problem 1.3 Slope Field

1.2 Analytic Technique: Separation of Variables 1.3 Qualitative Technique: Slope Fields

1.3 Slope Field 1.4 Numerical Technique

1.3 Qualitative Technique: Slope Fields 1.4 Numerical Technique: Euler’s Method

1.5 IVP解: 存在性與唯一性 1.6 質分析, phase line(for autonomous system)

1.5 Existence and Uniqueness of Solutions 1.6 Equilibria and the Phase Line

1.7 Bifurcation and one supplement for 1.6(real problem)

1.6 Equilibria and the Phase Line 1.7 Bifurcations

Complete Chapter 1

1.1 Modeling via Differential Equations

Chapter 2: 1st order system and 2nd order equation

2.1 Modeling via Systems

2.2 Geometry of the system(by vector field) 2.3 Damped harmonic oscillator

2.2 The Geometry of Systems 2.3 The Damped Harmonic Oscillator

2.4 Special systems(solutions can be solved explicitly) 2.5 Euler’s method for 1st order system

2.4 Additional Analytic Methods for Special Systems 2.5 Euler’s Method for Systems

2.6 存在與唯一性(for 1st order system) 2.7 (假的) 3D-system (流行病傳染問題) 2.8 (真的) 3D-system (氣象預報簡化系統: Lorenz equation)

2.6 Existence and Uniqueness for Systems 2.7 The SIR Model of an Epidemic 2.8 The Lorenz Equations

3.1 線性理論

3.1 Properties of Linear Systems and the Linearity Principle

3.2 Straight-Line Solutions 3.3 Phase Portraits for Linear Systems with Real Eigenvalues

3.2 Straight-Line Solutions 3.3 Phase Portraits for Linear Systems with Real Eigenvalues

3.2 Straight-Line Solutions 3.3 Phase Portraits for Linear Systems with Real Eigenvalues 3.4 complex valued eigenvalues

3.2 Straight-Line Solutions 3.3 Phase Portraits for Linear Systems with Real Eigenvalues 3.4 Complex Eigenvalues

3.5 Special cases of Linear systems

3.5 Special Cases: Repeated and Zero Eigenvalues

To complete 3.5

3.5 Special Cases: Repeated and Zero Eigenvalues

3.5 to complete 3.6 2nd order linear system and 2D linear system

3.5 Special Cases: Repeated and Zero Eigenvalues 3.6 Second-Order Linear Equations

To complete 3.6

3.6 Second-Order Linear Equations

3.7 Trace – Determinant Plane

3.7 The Trace-Determinant Plane

To complete chapter 3 3.7 Part II: T-D plane 3.8 3D linear system

3.7 The Trace-Determinant Plane 3.8 Linear Systems in Three Dimensions

Non-linear 2nd order ode:

Pendulum motion

Pendulum motion and spring motion

Differences of phase portraits between linear and Nonlinear 2nd order equations

Continue nonlinear 2nd order ode

Special case for solving problems

課程書目

Differential Equations by P. Blanchard, R. L. Devaney, G. R. Hall, 4th edition.

評分標準

項目

百分比

四次考試(最佳3次每次30%，剩餘1次10%)

100%

授課對象：大學2年級學生

預備知識：微積分