Find all Periodic function

Question

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!

Answer 1

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.