1. Ordinary differential equations: existence theorems, linear systems, Sturm-Liouville problems.

2. Partial differential equations. Classification of second order equations, characteristic surfaces, properties of elliptic, parabolic and hyperbolic equations including questions of existence, uniqueness, explicit study of representative or seminal simple cases.

Recommended Material:

1. Braun: Differential Equations and Their Applications, Springer 1976, especially Chapters 1, 3, and 5, but really the whole book.

2. Epstein: Partial Differential Equations, Krieger, Chapters 6, 8, 10 are most relevant but also good is the chapter on partial differential equations in Sokolnikoff and Redheffer or any one of the many applied mathematics compendiums ("Advanced Calculus for Engineers" or some such) treating the simplest cases of the most important classical problems of mathematical physics, mostly by the method of separation of variables or integral transforms.