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I need some help in the following problem. Let $a,b:\mathbb{R} \rightarrow \mathbb{R}$ be continuous functions with $b(x) \neq 0$. Suppose that $a, b$ are periodic functions with period $p>0$ and the differential equation $y’ + a(x)y = 0$ does not have periodic solutions with period $p$ (except the trivial solution $y(x) = 0$). Prove that the differential equation $y’ + a(x)y = b(x)$ has, at most, one periodic solution with period $p$. Thanks.

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closed as off-topic by MathOverview, Simply Beautiful Art, Xander Henderson, user99914, choco_addicted Oct 7 '17 at 5:00

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