SOS Mathematics

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
  • Radioactive Decay
  • Newton's Law of Cooling
  • Orthogonal Trajectories
  • 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
  • Method of Undetermined Coefficients
  • Method of Variation of Parameters
• Euler-Cauchy Equations
• Series Solutions
  • Introduction
  • Derivatives and Index Shifting
  • First Examples
  • Airy's Equation
  • The Radius of Convergence of Series Solutions
  • 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
  • Introduction and First Definitions
  • Vector Representations of Solutions of Linear Systems
  • Eigenvalues and Eigenvectors Technique
    • Real Eigenvalues
    • Repeated Eigenvalues
    • Complex Eigenvalues
    • Zero Eigenvalues
  • Qualitative Analysis of Linear Systems
• Nonlinear Systems
  • 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