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