Contents

1 Introduction

Learning about differential equations

Solving differential equations using Maxima

Solving differential equations using SageMath

2 Review

Complex numbers

Factoring

Hyperbolic functions

Calculus

Power series

Linear independence of functions

3 First order ODEs

Integrals as solutions

Slope fields

Separable equations

Linear equations

Substitution

Mixing problems and other applications

Autonomous equations

Euler's method

4 Linear algebra

Solving linear systems: Gauss’s method

Transition to matrix notation

Gauss-Jordan reduction

Solving systems of linear equations using Maxima

Solving systems of linear equations using SageMath

Matrix operations

Using Maxima to do linear algebra

Determinants

Eigenvalues and eigenvectors

5 Higher order ODEs

Second order linear ODEs

Constant coefficient second order linear ODEs

Reduction of order

Higher order linear ODEs

Mechanical vibrations

Nonhomogeneous equations: Undetermined coefficients

Nonhomogeneous equations: Variation of parameters

Forced oscillations and resonance

6 Systems of ODEs

Introduction to systems of ODEs

Linear systems of ODEs

Eigenmethod

Two dimensional systems and their vector fields

Second order systems and applications

7 Power series methods

Preliminaries

Solutions about ordinary points

Solutions about singular points and the method of Frobenius

8 Laplace transform method

Preliminaries

Transforms of derivatives and ODEs

Convolution

Dirac delta and impulse response

9 Useful formulas

Number sets

Exponentials and logarithms

Trigonometry

Hyperbolics

Antiderivatives

Taylor series