Introduction
Some basic mathematical models; Direction fields
Solutions of some differential equations
Classification of differential equations
First Order Differential Equations
Linear equations with variable coefficients
Separable equations
Modeling with first order equations
Differences between linear and nonlinear equations
Autonomous equations and population dynamics
Exact equations and integrating factors
Numerical Methods
The Euler or tangent liner method
Improvements on the Euler method
The Runge-Kutta method
Second Order Linear Equations
Homogeneous equations with constant coefficients
Fundamental solutions of linear Homogeneous equations
Linear independence and the Wronskian
Complex roots of the characteristic equation
Repeated roots; Reduction of order
Nonhomogeneous equations; Method of undetermined coefficients
Variation of parameters
Mechanical and electrical vibrations
Higher Order Linear Equations
General theory of n-th order linear equations
Homogeneous equations with constant coefficients
The method of undetermined coefficients
The method of variation of parameters
The Laplace Transform
Definition of the Laplace transform
Solution of initial value problem
Systems of First Order Linear Equations
Review of Matrices
Systems of linear algebraic equations; linear independence,
eigenvalues, eigenvectors
Basic theory of systems of first order linear equations
Homogeneous linear systems with constant coefficients
Complex eigenvalues
Fundamental matrices
Nonlinear Differential Equations and Stability
The phase plane; Linear systems
Autonomous systems and stability
Almost linear systems
Competing species
Predator-Prey equations
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