Review of Separation of variables. Laplace Equation: Potential theory, eigenfunction expansions, conformal methods, Fourier transforms. Heat Equation: Eigenfunction expansions, integral transforms, ...

Ordinary differential equations: review of elementary ODEs, initial and boundary value problems, Fredholm Alternative Theorem, Power series solution of ODEs, Special functions, Sturm-Liouville ...

Techniques for solving these for various initial and boundary value problems on bounded and unbounded domains, using eigenfunction expansions (separation of variables, and elementary Fourier series), ...