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I want to know of some sources where I can learn the methods to solve equations like this:

$-d^2f/dx^2 = e^x$ on [0,2pi], $f \in V$ = {periodic, $C^{\infty} \cap L^2$}, $f(0) = 1$

The prof. talked specifically about using eigenvalues and fourier series.

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You may want to check your notation again. – Ron Gordon Feb 3 '13 at 19:27
I forgot the 'f' in the beginning... other than that this is verbatim what he wrote... I think I am supposed to use Fourier transform. – meeka Feb 3 '13 at 20:58
I'm not sure how the equation above produces anything periodic. To solve the above, you simply integrate twice to get $f(x) = a + b x - e^x$. You have one boundary condition, which still leaves you with a free parameter. – Ron Gordon Feb 3 '13 at 21:49

The approach your Pofessor was talking is called Galerkin approximations and more specifically Spectral methods (which use fourier basis functions). You can check in wiki there are some literature, I would suggest you to start with "Chebyshev and Fourier Spectral Methods" by John P. Boyd. - it is very nice for the beginning.

Small note regarding your problem. You can look for solution of your problem in the form: $$f(x)=\sum_{i=1}^{\infty} (a_isin(ix)+b_icos(ix))$$ both $sin(ix)$ and $cos(ix)$ are eigenfunctions of the operator of your problem and they are periodic on the interval $[0;2\pi]$.

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