Green's Functions Vs. Fourier Series for ODE's I'm kind of confused about when you'd use a Green's function as opposed to using a Fourier series when solving a non-homogeneous second order ode? What is the rule of thumb one should follow, when does one work & the other not work? As far as I can see they both only come up when the forcing term is either periodic or piecewise, but I'm not sure about that! Any help is welcome! :)

The most important difference is that these methods achieve different results: you get a solution as an integral with Green's function, or as a series with Fourier series. So it makes sense to have both methods, and maybe even to use both on the same problem when practical. 
I would consider using Fourier series when the forcing term suggests that I may be able to compute the coefficients of its Fourier series. Green's function approach is less sensitive to the form the forcing term: once you found Green's function, you have the  solution  or every forcing term $f$ as an integral involving $f$: something like $u(x)=\int G(x,y)f(y)$. At worst, the integral can be evaluated numerically for several values of $x$, and an approximate plot of $x$ can be obtained by interpolation ("connecting the dots").
Then again, with a Fourier series approach you can also compute several coefficients $c_n$ of the Fourier series of $f$ numerically. This may be enough to obtain an explicit trigonometric polynomial (partial sum of Fourier series) that gives a decent approximation to the exact solution. This is something you don't get from "connecting the dots" with the Green's function approach.
