Mathematics 215 (Differential Equations I), Spring 2001

• Introduction; first order equations (Sections 2.1 - 2.8; 8 hours)
  • Direction fields
  • Linear equations
  • Separable equations
  • Exact equations and integrating factors (may be delayed until after "Numerical methods")
  • Applications: mixing, falling bodies, population dynamics
• Numerical methods (Sections 8.1 - 8.4, 8.6; 5 hours)
  • The Euler method
  • Errors and extrapolation
  • Improved Euler method
  • Fourth-Order Runge-Kutta method
  • Adjustment of step size
• Second order linear equations (Sections 3.1 - 3.9; 7 hours)
  • Constant-coefficient equations
  • Linearity and its consequences
  • Complex roots
  • Higher order equations (includes some topics from Sec. 4.1-4.2)
  • Method of undetermined coefficients
  • Non-constant coefficients: variation of parameters
  • Applications: mechanical vibrations, resonance
• First order constant-coefficient linear systems (Sections 7.1, 7.4 - 7.9, 9.1; 6 hours)
  • Eigenvalues and eigenvectors
  • Fundamental matrices
  • Phase portraits for two-dimensional systems
• Two-dimensional nonlinear systems (Sections 9.2 - 9.5, 9.8; 7 hours)
  • Critical points, linearization, and stability
  • Sketching the phase portrait
  • Applications: predator and prey, damped pendulum
  • Chaos (if time permits)