Ordinary Differential Equations

• Definition, reduction of explicit equations to first order
• Existence and uniqueness of solutions
• Differential inequalities
• Solutions to specific equations
  • One-dimensional time-independent equations of first or second order

This book aims to lead the reader through the topic of differential equations, a vital area of modern mathematics and science. This book provides information about the whole area of differential equations, concentrating first on the simpler equations.

• Introduction
• Preliminaries from calculus
• 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
  • The Picard–Lindelöf theorem
  • Peano's theorem
  • Blow-ups and moving to boundary
  • 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

Differential Equations and Boundary Value Problems- C.H. Edwards Jr and David E. Penny
MIT Open Courseware- http://ocw.mit.edu/index.html

• Kong, Qingkai (0000). A Short Course in Ordinary Differential Equations. Universe: Publisher.
• Walter, Wolfgang (1998). Ordinary Differential Equations. New York: Springer.