Classification of second-order PDEs with constant coefficients; boundary value problems on an interval; Robin, Dirichlet, and von Neumann conditions; existence and uniqueness; separation of variables; eigenvalue problems and eigenfunction expansion; Fourier series and applications; inhomogenous problems; integral transform methods; the fundamental solution; Green functions; maximum-minimum principles; the method of characteristics; parabolic equations and the heat equation; elliptic equations and the Laplace equation; the steady-state and equilibrium problems for elliptic equations; the energy integral for the Laplace equation; the Dirichlet problem for the Laplace equation; wave propagation; the d'Alembert solution for hyperbolic equations; the Cauchy problem for hyperbolic equations; the method of characteristics.