First-order partial differential equations: Method of characteristics, Cauchy
problem, Hamilton-Jacobi equations (including either the Hopf-Lax formula or
introduction to viscosity solutions, as per audience interest).
Second order partial differential equations: The Cauchy problem and
classification of second order equations, Holmgren’s uniqueness theorem;
Laplace equation, Heat equation, Wave equation, Some methods of solutions,
Variables separable method.
Brief introduction to the weak formulation (time permitting): Motivations,
Dirichlet functional.
Recommended Books

John, F., Partial Differential Equations, Springer (International Students
Edition), 1971.

Evans, L. C., Partial Differential Equations, AMS, 1998.