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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
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No, it doesn't. But in in special cases it has at least one periodic solution. The equation is known as Hill's equation, and the theory of its solutions, whether periodic or not, is known as Floquet theory. The referenced wikipedia page deals with first order systems; you'll have to rewrite your second order equation to a first order system to use the theory directly.

If you want to learn more, the little book Hill's equation by Magnus and Winkler (in the Dover series) is an excellent resource.

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

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