Fall 2021 lectures with steve butler

Introduction to differential equations -- Springing into action

Verifying solutions; Initial value problems; Mathematical modeling

Setting up a differential equation; Terminology; Examples of things that can go awry

Simple physics examples; Existence and uniqueness

Slope fields; Separable differential equations

Practice with separable differential equations

Method of integrating factor

Substitution method

Exact ODEs; Population models

Population models; Autonomous differential equations

Numerical approximation (Euler)

[Optional topic; not covered on exam/homework]

nth-order differential equations; Wronskian and linear independence

Solving constant coefficient homogeneous linear differential equations (1/2)

Solving constant coefficient homogeneous linear differential equations (2/2)

Nonhomogeneous strategy; Spring-mass systems

Remaining material on spring-mass; short review of material

Exam 1 practice / solutions

Method of undetermined coefficients

Method of variation of parameters

Practice with variation of parameters; introduction to systems of differential equations

Substitution and elimination to solve systems of differential equations

Basics of matrices and linear algebra

Basics of differential equations for systems

Practice with systems of differential equations

Eigenvalues and eigenvectors

Complex eigenvalues; direction fields

Handling missing eigenvectors (how to "bump" for systems)

Fundamental matrix and matrix exponentials

Comments on nonhomogeneous ODE systems [optional material for Fall 2021]

Introduction to the Laplace transform

Laplace transforms and derivatives

Laplace transforms and partial fractions

More rules for Laplace transforms

The 𝛿-function and dealing with shifts

Additional practice with Laplace transforms

Review of material for midterm 2

Exam 2 practice / solutions

Introduction to using power series to solve differential equations

Radius of convergence of solutions

Ordinary/singular points

Final exam practice / solutions