S.O.S. Mathematics

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Introduction and First Definitions

Modeling via Differential Equations

First Order Differential Equations

* Linear Equations
* Separable Equations
* Qualitative Technique: Slope Fields
* Equilibria and the Phase Line
* Bifurcations
* Bernoulli Equations
* Riccati Equations
* Homogeneous Equations
* Exact and Non-Exact Equations
* Integrating Factor technique
* Some Applications
o Radioactive Decay
o Newton's Law of Cooling
o Orthogonal Trajectories
o Population Dynamics
* Numerical Technique: Euler's Method
* Existence and Uniqueness of Solutions
* Picard Iterative Process

Second Order Differential Equations

* Nonlinear Equations
* Linear Equations
* Homogeneous Linear Equations
* Linear Independence and the Wronskian
* Reduction of Order
* Homogeneous Equations with Constant Coefficients
* Non-Homogeneous Linear Equations
o Method of Undetermined Coefficients
o Method of Variation of Parameters
* Euler-Cauchy Equations
* Series Solutions
o Introduction
o Derivatives and Index Shifting
o First Examples
o Airy's Equation
o The Radius of Convergence of Series Solutions
o Hermite's Equation

Higher Order Linear Equations

* Introduction and Basic Results
* Homogeneous Linear Equations with Constant Coefficients
* Non-Homogeneous Linear Equations
* Method of Undetermined Coefficients
* Method of Variation of Parameters

Laplace Transform

* Basic Definitions and Results
* Application to Differential Equations
* Impulse Functions: Dirac Function
* Convolution Product
* Table of Laplace Transforms

Systems of Differential Equations

* Introduction and Motivation
* Second Order Equations and Systems
* Euler's Method for Systems
* Qualitative Analysis
* Linear Systems
o Introduction and First Definitions
o Vector Representations of Solutions of Linear Systems
o Eigenvalues and Eigenvectors Technique
Real Eigenvalues
Repeated Eigenvalues
Complex Eigenvalues
Zero Eigenvalues
o Qualitative Analysis of Linear Systems
* Nonlinear Systems
o Equilibrium Point Analysis: Linearization Technique

Fourier Series

* Fourier Series: Basic Results
* Fourier Sine and Cosine Series
* Convergence of Fourier Series
* More on Convergence of Fourier Series
* Gibbs Phenomenon
* Bessel's Inequality and Parseval Formula: The Energy Theorem
* Operations on Fourier Series
* Application of Fourier Series to Differential Equations

APPENDIX

* Tableshttp://www.sosmath.com/index.htmlhttp://www.sosmath.com/index.html