Ordinary Differential Equations

Ordinary Differential Equations

covering uses of and solutions to ordinary differential equations

The Rössler Attractor. This chaotic system is generated by a system of ordinary differential equations.

This book aims to lead the reader through the topic of differential equations, a vital area of modern mathematics and science. It is hoped that this book will provide information about the whole area of differential equations, but for the moment it will concentrate on the simpler equations.

Table of contents

• Introduction
• Form and Solutions of Differential Equations
• First-Order Differential Equations
  • Separation of Variables
  • Linear Differential Equations
  • Exact Differential Equations
  • Substitution Methods
  • Bernoulli Equations
  • Ricatti Equations
  • Orthogonal and Oblique Trajectories
  • Equations of higher degrees
  • Equations without x or y
  • Equations that are homogeneous in x and y
  • d'Alembert's Equation
  • Clairaut Equations
  • Legendre Transformations
  • Graphing Differential Equations
• Second-Order Differential Equations
  • Constant Coefficients
  • Series Solutions
  • Hypergeometric Equation
  • Frobenius Solution to the Hypergeometric Equation
  • Legendre Equation
  • Bessel Equation
  • Mathieu Equation
  • Continued Fraction Solutions
  • Applications of Second-Order Differential Equations
• Higher Order Differential Equations
  • Linear equations
    • General Linear Equations
    • Infinite Series Solutions to Linear Equations
  • Integration methods
    • Laplace Transform
    • Bessel Function
    • Euler Transform
    • Mellin Transform
    • Hypergeometric Series
    • Double Integration
• Sturm-Liouville theory
• Systems of linear differential equations
• Nonlinear Systems
• Green's Functions
• Existence and Uniqueness of Solutions
  • Global uniqueness of solution over interval
  • Maximum domain of solution
  • The Successive Approximations Method of Proof
  • Applications to Linear Equations
  • The Cauchy-Lipschitz Method of Proof
  • Existence Theorems for Complex Numbers
• Continuous Transformation Groups
  • Infinitesimal Transformations
  • Invariant Functions
• Glossary
• List of Some Equations
• Help Needed
• Roadmap

Sources

Differential Equations and Boundary Value Problems- C.H. Edwards Jr and David E. Penny

MIT Open Courseware- http://ocw.mit.edu/index.html