Differential equations

First order differential equations. Existence and uniqueness of solutions: initial value problems. Linear versus nonlinear equations. Integrating factor method and Separations of Variables.

Second order linear differential equations. The superposition principle, general solution, linear (in)dependence, fundamental set of solutions, Wronskian, Abel's Theorem. Order reduction, variation of parameters.

Integral transforms, the Laplace transform, solving linear differential equations with discontinuous forcing.

Systems of differential equations, both linear and locally linear. Phase plane analysis.

Linear partial differential equations. Method of separation of variables, including Fourier series. Boundary value problems. Selected applications, including diffusion, wave and Laplace equations.