Department of Mathematics

Van Vleck Hall, 480 Lincoln Drive, Madison, WI

Math 319: Techniques in Ordinary Differential Equations

Student Body: 

Students in science, mathematics, computer sciences and engineering

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.


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: 


Course Content: 
  • 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 and other numerical methods (optional)
  • 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
  • Series solutions of linear equations
    • Review of power series, power series solutions
    • Euler equations, Solutions at a regular singular point (optional)
  • 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
  • Boundary value problems (time permitting)
    • physical origins via separation of variables from PDE
    • Fourier expansions
    • eigenvalue problems
    • more general expansion methods
  • Two dimensional systems and the phase plane (time permitting)
    • classification of (equilibria for) linear systems
    • qualitative behavior of nonlinear systems: classification of equilibria; stability
    • applications, e.g. to the pendulum, population models
  • 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
3. (N-A)
Math 222

UW-Madison Department of Mathematics
Van Vleck Hall
480 Lincoln Drive
Madison, Wi  53706

(608) 263-3054

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