**This course was offered during the Spring 2015 semester at Harvard University by Professor Michael Brenner and Dr. Sabetta Matsumoto.**

Section 1 - Constructing exact solutions for first-order ODE's; solution for ODE's using dominant balance; using MATLAB to see how solutions behave.

Section 2 - Exact solutions for linear second-order ODE's; linear dependence/independence; more dominant balance.

Section 3 - Method of undetermined coefficients solving for in-homogeneous second order differential equations; characterizing qualitative behavior of ODE through fixed points and phase planes.

Section 4 - Finding fixed points; characterizing stability; determining nullclines; and piecing everything together to sketch dynamics in phase plane.

Section 5 - Using power series approach to solve ODE's with variable coefficients; solve an eigenvalue problem; learn to expand in Fourier series.

Section 6 - Convergence of Fourier series and the Gibbs phenomenon; solving Sturm-Liouville problem.

Section 8 - Solving Laplace equation; deriving the diffusion equation from random walkers; solving the diffusion equation.

Section 9 - Solving diffusion equation with in-homogeneous boundary conditions; dealing with diffusion equation in polar/spherical coordinates; solving the wave equation.

Section 10 - Thinking qualitative about solutions to PDE's combining: ODE's with advection, ODE's with diffusion, and advection with diffusion; dealing with separation of variables in three dimensions.