Students in science, mathematics, computer sciences and engineering

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.

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- 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