## Syllabus for the Comprehensive Exam in Differential Equations

### Part I: Partial Differential Equations

1. Basic Material: Motivation and derivation of basic PDE; initial and boundary value problems; existence and uniqueness; classification of first- and second-order equations
2. First-order Equations: Method of characteristics; transport equations; the Hamilton-Jacobi equations; conservation laws; weak solutions; solutions with shocks
3. The Laplace and Poisson Equations: Maximum principles; mean value properties; regularity; Green's functions
4. The Heat Equation: The fundamental solution; maximum principles; regularity; energy methods
5. The Wave Equation: D'Alembert's formula; Kirchoff's and Poisson's formulas; domain of dependence; finite speed of propagation; Huygens' principle; energy methods
6. Existence and Uniqueness: Well-posed PDE; characteristic and non-characteristic surfaces; power series; the Cauchy-Kowalevski and Holmgren theorems
7. Techniques: Separation of variables and eigenfunction expansion; applications of the Fourier transform to PDE

### Part II: Ordinary Differential Equations

1. General Properties: Existence; uniqueness; dependence on parameters
2. Asymptotic Properties: Stability; Lyapunov functions; attractors; limit sets; chain recurrence
3. Poincaré-Bendixon Theory
4. Linear Systems: Floquet theory; stability; nonlinear perturbations
5. Mappings: Return mappings and time-t mappings
6. Local Equilibria: The Hartman-Grobman theorem; stable/unstable/center manifolds; elementary local bifurcations

Suggested textbooks: Partial Differential Equations by Evans; Introduction to Partial Differential Equations by Folland; Ordinary Differential Equations with Applications by Chicone; Ordinary Differential Equations by Hale
Suggested courses: 6307 and 6341