Ordinary Differential Equations

Ordinary differential equations are equations involving derivatives in one direction, to be solved for a solution curve.

Table of contents

• Introduction

Existence of ODEs

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

First order equations

• One-dimensional first-order linear equations
• Separable equations
• Integrating factor
• Exact equations
• Linearize
• Autonomous equations
• Derived cases 1: Rational functions in the right hand side
• Derived cases 2: Bernoulli equations, Ricatti equations
• Derived cases 3: Euler factors

Second order equations

• Homogeneous second order equations
• Nonhomogeneous second order equations:Method of undetermined coefficients
• Nonhomogeneous second order equations: Variations of parameters
• Nonhomogeneous second order equations: Reduction of order

Higher order equations

• Homogeneous higher order equations
• Nonhomogeneous higher order equations
• Linear autonomous equations of higher order with constant coefficients
• Linear autonomous equations of higher order with varying coefficients

Systems of equations

• Homogeneous linear systems with constant coefficients
• Nonhomogeneous linear systems: Diagonalization method
• Nonhomogeneous linear systems: Method of undetermined coefficients
• Nonhomogeneous linear systems: Integrating factor
• Nonhomogeneous linear systems: Variation of parameters

Laplace Transform

• Laplace transform for 1D ODEs
• Laplace transform for systems
• Lerch's theorem proof

Nonlinear systems of equations

• Autonomous systems
• Locally linear

Lyapunov's stability results

• Lyapunov's first method
• Lyapunov's second method

Sources

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.