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BrainHurts
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Homework Statement
Consider the first-order non-autonomous equation ##x' = p(t) x##, where ##p(t) ## is differentiable and periodic with period ##T##. Prove that all solutions of this equation are period with period ##T## if and only if ##\int_0^T p(s) ds = 0##.
Homework Equations
The Attempt at a Solution
I'm not quite sure, first I got that the general solution to ##x' = p(t) x ## is
##x(t) = C e^{\int p(t) dt}##. Now I know that ##p(t+T) = p(t)##.
So we have ##x' = p(t) x## ##(\star) ##, ##p## is periodic with period ##T## and ##x(t) = C e^{\int p(t) dt}##
Any hints on how to proceed?