Differential Equations Notes

Basics:

• Basic Concepts

• Definitions

• Direction Fields

First Order Differential Equations:

• First Order Differential Equations

• Linear Equations

• Separable Equations

• Exact Equations

• Bernoulli Differential Equations

• Substitutions

• Intervals of Validity

• Modeling with First Order Differential Equations

• Equilibrium Solutions

• Euler’s Method

Second Order Differential Equations:

• Second Order Differential Equations

• Basic Concepts

• Real Roots

• Complex Roots

• Repeated Roots

• Reduction of Order

• Fundamental Sets of Solutions

• More on the Wronskian

• Nonhomogeneous Differential Equations

• Undetermined Coefficients

• Variation of Parameters

• Mechanical Vibrations

Laplace Transforms:

• Laplace Transforms

• The Definition

• Laplace Transforms

• Inverse Laplace Transforms

• Step Functions

• Solving Initial Value Problems with Laplace Transforms

• Nonconstant Coefficient IVP’s

• Initial Value Problems with Step Functions

• Dirac Delta Function

• Convolution Integral

• Table of Laplace Transforms

Systems of Differential Equations:

• Systems of Differential Equations

• Review : Systems of Equations

• Review : Matrices and Vectors

• Review : Eigenvalues and Eigenvectors

• Systems of Differential Equations

• Solutions to Systems

• Phase Plane

• Real Eigenvalues

• Complex Eigenvalues

• Repeated Eigenvalues

• Nonhomogeneous Systems

• Laplace Transforms

• Modeling

• Series Solutions

• Review : Power Series

• Review : Taylor Series

• Series Solutions

• Euler Equations

Higher Order Differential Equations:

• Higher Order Differential Equations

• Basic Concepts for nth Order Linear Equations

• Linear Homogeneous Differential Equations

• Undetermined Coefficients

• Variation of Parameters

• Laplace Transforms

• Systems of Differential Equations

• Series Solutions