I'm a bit stuck on a certain review question for ODEs. Here's how it goes:
Given the one dimensional equation $y\prime(x) = ay+b(x)$, with $a\neq 0$ and $b:\mathbb R\to \mathbb R$ continuous and $T$-periodic, prove that there exists a unique $T$-periodic solution.
Now, I know that the flux of this particular equation can be explicitly written as: $$\varphi(x;x_0,k) = ke^{a(x-x_0)} + \int_{x_0}^x{b(s)e^{a(x-s)}ds}$$
(This is, each solution to the IVP $y(x_0) = k$)
So I figure I can prove existence by choosing adequate $x_0$ and $k$ (my intuition indicates $k = 0$), such that I can prove periodicity, and uniqueness will be "for-free" by Picard's theorem. I've tried writing the above expression at $x$ and $x+T$ and equating, but I haven't gotten anywhere really. Maybe this isn't the right approach anyway. Can someone help?
hint: because you are looking for periodic solution you may as well take $x_0 = 0.$ the integral representation of the solution then $$ y(T) = y(0) = y(0)e^{aT} +\int_0^T b(s)e^{a(T-s)} \ ds $$ which gives you the constraint $$(e^{-aT}-1)y(0) = \int_0^T b(s)e^{-as} \ ds $$