Math 319 - Techniques in Ordinary Differential Equations

• Prerequisites: Math 222
• Frequency: Fall (I), Spring (II), Summer (SS)
• Student Body: Students in science, mathematics, computer sciences and engineering
• Credits: 3. (N-A)
• Recent Texts: Elementary Differential Equations, Boyce and DiPrima.
• Course Coordinator: Paul Rabinowitz
• Background and Goals: This course presents techniques for solving and approximating solutions to ordinary differential equations. It is primarily for students in disciplines which emphasize methods. Math 319 is a prerequisite for Math 519, an advanced course intended for math majors and others who need a theoretical background in ordinary differential equations or a more detailed study of systems and/or behaviour of solutions.
• Alternatives: Math 320 covers linear algebra together with differential equations but it covers linear systems of differential equations and initial value problems only. Math 319 is a more extensive study of the subject.
• Subsequent Courses: N/A

Content coverage:

• Introduction: definition of an ODE, basic problems (IVP and BVP), examples
• First order equations
  • linear: homogeneous and inhomogeneous
  • nonlinear: separable
  • direction fields
  • the basic existence and uniqueness theorem (for first order equations)
  • the Euler scheme
  • other numerical methods
• Second order linear equations with constant coefficients
  • homogeneous case
  • inhomogeneous equations via methods of annihilators and variation of parameters
  • remarks on higher order equations, linear independence, and the Wronskian
  • applications to forced oscillation problems, effect of resonances
• Laplace transform
  • definition and elementary properties
  • application to constant coefficient linear equations
  • discontinuous forcing terms
• First order systems
  • conversion of 2nd and higher order equations to systems (focusing on systems in the plane and simple cases in 3 dimensions)
  • discussion of algebraic properties of vectors in and matrices on the plane and 3 dimensional space. Also differentiation of vector and matrix functions
  • solution of linear constant coefficient systems
• Two dimensional systems and the phase plane
  • classification of (equilibria for) linear systems
  • qualitative behavior of nonlinear systems: classification of equilibria; stability
  • applications, e.g. to the pendulum, population models
• Boundary value problems
  • physical origins via separation of variables from PDE
  • Fourier expansions
  • eigenvalue problems
  • more general expansion methods
• More on systems (time permitting)
  • qualitiative behavior in the phase plane: limit cycles, heteroclinics, homoclinics, etc.; the Poincare-Bendixson theorem
  • the dependence of equations on parameters; bifurcation
  • chaotic solutions
• Series methods (time permitting)