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Differential Equations

微分方程

本課程是由 國立陽明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.

課程用書：

Differential Equations by P. Blanchard, R. L. Devaney, G. R. Hall, 4th edition Publisher: Cengage Learning

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授課教師 應用數學系 李榮耀老師
課程學分 3學分
授課年度 103學年度
授課對象 大學二年級學生
預備知識 微積分
課程提供 課程影音   課程綱要   課程行事曆

課程目標

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

章節 章節內容
Introduction of ODEIntroduction of ODE
Big Picture for ODE1.8 Linear Equations
1st order linear system1.8 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 coefficeints3.6 Second-Order Linear Equations
Practical Problem1.1 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 Picture1.2 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 11.1 Modeling via Differential Equations
Chapter 2: 1st order system and 2nd order equation2.1 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 systems3.5 Special Cases: Repeated and Zero Eigenvalues
To complete 3.53.5 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
To complete 3.63.6 Second-Order Linear Equations
3.7 Trace – Determinant Plane3.7 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%

本課程行事曆提供課程進度與考試資訊參考。

授課日期
授課主題
參考課程進度

2014/09/15

Introduction of ODE
  • Introduction of ODE
2014/09/17Big Picture for ODE
  • 1.8 Linear Equations
2014/09/221st order linear system
  • 1.8 Linear Equations
    1.9 Integrating Factors for Linear Equations
2014/09/241st order linear system: 量的分析
  • 3.6 Second-Order Linear Equations
    4.1 Forced Harmonic Oscillators
    4.2 Sinusoidal Forcing
2014/09/292nd order linear equation with constant coefficeints
  • 3.6 Second-Order Linear Equations
2014/10/01Practical Problem
  • 1.1 Modeling via Differential Equations
2014/10/06Two more “derivation” of math models from:
1. Predator-Prey problem
2. Deposit and Withdraw problem
  • 2.1 Modeling via Systems
2014/10/081.2 ODE Big Picture
  • 1.2 Analytic Technique: Separation of Variables
2014/10/131.2 Mixed Problem
1.3 Slope Field
  • 1.2 Analytic Technique: Separation of Variables
    1.3 Qualitative Technique: Slope Fields
2014/10/15期中考
2014/10/201.3 Slope Field
1.4 Numerical Technique
  • 1.3 Qualitative Technique: Slope Fields
    1.4 Numerical Technique: Euler’s Method
2014/10/221.5 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
2014/10/271.7 Bifurcation and one supplement for 1.6(real problem)
  • 1.6 Equilibria and the Phase Line
    1.7 Bifurcations
2014/10/29Complete Chapter 1
  • 1.1 Modeling via Differential Equations
2014/11/03Chapter 2: 1st order system and 2nd order equation
  • 2.1 Modeling via Systems
2014/11/052.2 Geometry of the system(by vector field)
2.3 Damped harmonic oscillator
  • 2.2 The Geometry of Systems
    2.3 The Damped Harmonic Oscillator
2014/11/102.4 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
2014/11/122.6 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
2014/11/173.1 線性理論
  • 3.1 Properties of Linear Systems and the Linearity Principle
2014/11/19期中考
2014/11/243.2 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
2014/11/263.2 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
2014/12/013.5 Special cases of Linear systems
  • 3.5 Special Cases: Repeated and Zero Eigenvalues
2014/12/03運動會
2014/12/08To complete 3.5
  • 3.5 Special Cases: Repeated and Zero Eigenvalues
2014/12/103.5 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
2014/12/15To complete 3.6
  • 3.6 Second-Order Linear Equations
2014/12/17期中考
2014/12/223.7 Trace – Determinant Plane
  • 3.7 The Trace-Determinant Plane
2014/12/24To 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
2014/12/29Non-linear 2nd order ode:
1. Pendulum motion
2. Pendulum motion and spring motion
3. Differences of phase portraits between linear and Nonlinear 2nd order equations		 
2014/12/31Continue nonlinear 2nd order ode 
2014/01/05Special case for solving problems 
2014/01/07期末考
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