MAT267F Ordinary Differential Equations

Preliminary Course Outline:

1. Introduction
   • Types of differential equations
   • Direction fields for first order ODE's, isoclines
2. First order ODE's
   • Separable ODE's
   • Homogeneous ODE's
   • Linear first order ODE's: Integrating factor
   • ODE's corresponding to exact differential 1-forms
   • Some other types of explicitly solvable ODE's
   • The existence and uniqueness theorem for first order ODE's: Picard iteration, uniform convergence.
3. Linear n-th order ODE's
   • Existence and uniqueness theorem
   • Fundamental systems of solutions, Wronskians
   • Constant coefficients
   • Complex roots
   • Repeated roots, reduction of order
   • Inhomogeneous equations: Undetermined coefficients, variation of parameters
   • Applications: Oscillations, vibrations
4. Systems of first order ODE's
   • Relation with higher order ODE's
   • Existence and uniqueness
   • Constant coefficient systems
   • Complex eigenvalues, repeated eigenvalues
   • Fundamental matrices
   • Inhomogeneous systems
5. Nonlinear Differential Equations and Stability
   • Phase portraits
   • Autonomous systems and stability
   • Linearizations
   • Applications
6. Series Solutions of Second Order Linear Equations
   • Review of power series
   • Series solution near ordinary point
   • Examples
7. Sturm oscillation theorem (?)
8. Numerical methods (?)