# Periodic solution of differential equation

let be the ODE $-y''(x)+f(x)y(x)=0$

if the function $f(x+T)=f(x)$ is PERIODIC does it mean that the ODE has only periodic solutions ?

if all the solutions are periodic , then can all be determined by Fourier series ??

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Have a look at Floquet theory. – J. M. Aug 8 '12 at 8:34

No, it doesn't mean that. For instance, $f(x)=0$ is periodic with any period, but $y''(x)=0$ has non-periodic solutions $y(x)=ax+b$.