20599 Ordinary Differential Equations II 1

Credits: 4 advanced credits in Mathematics

Prerequisites: Students must fulfill all English requirements and take bibliographic instruction in the Library.

Required: Infinitesimal Calculus I, Infinitesimal Calculus II, Infinitesimal Calculus III, Linear Algebra I, Ordinary Differential Equations I, Complex Functions

Recommended: Linear Algebra II

The course, based on Ordinary Differential Equations with Applications, by S.-B. Hsu (World Scientific, 2006), was developed by Eli Levin, Mireille Avigal and Zippy Berger.

Chapters: (1) Introduction; (2) Fundamental theory: Local existence and uniqueness of solutions of IVP, continuation of solutions, continuous dependence properties, differentiability of initial conditions and parameters, differential inequalities; (3) Linear systems: Fundamental matrices, linear systems with constant coefficients, two-dimensional linear autonomous systems, linear systems with periodic coefficients, adjoint systems; (4) Stability of nonlinear systems: Definitions, linearization, orbital stability; (5) Method of Lyapunov functions: An introduction to dynamical systems, Lyapunov functions, simple oscillatory phenomena; (6) Two-dimensional systems: Poincaré-Bendixson theorem; (7) Second order linear equations: Sturm's comparison and Sturm-Liouville boundary value problems, distributions, Green's function, Fredholm alternative for 2nd order linear equations; Additional topic Series solutions of second order linear differential equations: Series solution near an ordinary point and near a regular singular point, method of Frobenius, a regular singular point at infinity.

1A short non-credit seminar may be added to this course. See note under Mathematics seminars.