# 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: Gaussâ€™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