Find all Periodic function Can you help me with one question?
Find all twice continuously differentiable $2\pi$ periodic functions which:
$e^{ix} f''(x)+5f'(x)+f(x)=0$
probably has something to do with Fourier series
Any ideas?
Thanks!
 A: Such a function can be written in the form
$$f(t)=\sum_k c_ke^{-ikt}\ ,\tag{1}$$
where the summation ranges over ${\mathbb Z}$. Formally one then  has
$$f'(t)=\sum_k c_k\>(-ik) e^{-ikt},\qquad f''(t)=\sum_k c_k\>(-k^2)\>e^{-ikt}\ ,$$
$$e^{it}f''(t)=-\sum_k c_k\>k^2\>e^{-i(k-1)t}=-\sum_k c_{k+1}\>(k+1)^2\>e^{-ikt}\ .$$
The given ODE then implies
$$\sum_k e^{-ikt}\bigl(-(k+1)^2 c_{k+1}+c_k(1-5ik)\bigr)=0\ .$$
This leads to the recursion
$$c_k={(k+1)^2\over 1-5ik}c_{k+1}\qquad(k\in{\mathbb Z})\ .\tag{2}$$
Putting $k:=-1$ we obtain $c_{-1}=0$, and proceeding to the left we see that $c_k=0$ for all negative $k$.
We therefore rewrite $(2)$ in the form
$$c_{k+1}={1-5ik\over(k+1)^2}\>c_k\qquad(k\geq0)\ ,\tag{3}$$
whereby $c_0$ is arbitrary. The $c_k$ $(k\geq0)$ obtained in this way decrease fast enough to make our "Ansatz" $(1)$ not only formally, but also analytically valid. Since $c_0$ is arbitrary the recursion $(3)$ produces a one-dimensional vector space of $2\pi$-periodic solutions of the given ODE.
