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 with Boundary Value Problems 9E By Boyce & DiPrima
Course Coordinator:
none 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 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