WS-ODE Script

Introduction

• Autonomous Differential equation: $f$ is independent of $t$
• Rheonomous Differential equation: $f$ is dependent on $t$
• Expressing higher order ODE’s as a system of First order ODE’s
• Fundamental Solution

Solution Techniques

• Separating Variables
• Differential Equations in Homogeneous Variables
• Linear Equations of First Order - Variation of Constants
  • Solution Method for Variation of Constants
• Exact Differential Equation
  • Solution Method for Exact or Total Differential Equation
  • Solution Method to build a Potential function from a given Exact Differential Equation
• Bernoulli Differential Equation
• Euler Differential Equation

Existence and Uniqueness

• Lipschitz Continuity -SECOND DEFINITION SEEMS TO BE WRONG
  • Supremum Principle
• Uniqueness Theorem of Picard and Lindelof

Linear differential equations of $n^{th}$ Order

• Definition of a Linear Differential Equation
• Conditions for a Differential Equation to be Linear
• Linearly Independent Solutions
• Steps to check if a Solution is a Fundamental System of an ODE
  • Complex and Real Fundamental System
  • Variation of Constant for a Fundamental System
• Linearly Dependent and Independent Solutions
  • Homogeneous Solutions
  • Inhomogeneous Solutions

Linear Differential Equation with Constant Coefficients

• Complex Zeros
• One-Mass Oscillator

Systems of Linear Differential Equations

• General form of an Linear Differential Equation in Matrix form
• e-Ansatz for a Differential Equation in Matrix form
• General Solution
• solution method
• Variation of Constants

Laplace Transform

• Laplace Transform
• Basic Properties
  • Multiplication, Differentiation and Damping Theorem