I am preparing for my differential equations exam, and in the sample paper, which I have been provided with, there is only one problem, which is left for me to do. Its statement is as follows:
Let $q(x) \in \mathcal{C}[\mathbb{R}],~ q(x)\in\mathcal{T}[\mathbb{R}]~:~\exists T~\forall x \in \mathcal{D}_{q}~:~q(x+T) = q(x)$ ($q(x)$ is periodic with the least period of $T$).
Now, say $\exists ~y(x)$ being a non-trivial solution of $y''(x) + q(x)y(x) = 0 ~[~y(0)=y(T) = 0 \text{ by given }]$.
Prove that for some constant $C$ : $y(x + T) = Cy(x)$.
My attempt
What I first recognized is the Sturm-Liouville appearance of the ODE which I have been provided with. Since we know that $y(0) = y(T) = 0$, we may conclude that there are at least two zeroes on $\mathbb{R}$ for $y(x)$, now, we have to somehow elaborate on the infinity of the number of roots of the $y(x)$ solution. I guess, this may involve the Sturm comparison theorem, taking the given as base, but I don't know what should I compare $q(x)$ to? I mean, it is clearly periodic, and may have infinitely many roots on $\mathbb{R}$, but we actually know nothing about it. It may wholly lie above the $x$ axis, so that its periods don't imply any solutions for $x$. I guess, I have to somehow show that periodicity of $y$ is dependent on the periodicity of $q$, yet, I don't understand how to explicitly show this at all.
I am going to edit this post, if I come to any new ideas, which may be useful.
I would appreciate any help or reference to a book or solution of the similar-like problem.
Thank you a lot in advance!
Suppose the solution has $y'(T)=Cy'(0)$. If $C\neq 0$, show that $z(x)=y(x-T)/C$ also solves the same IVP (the same ODE with the same initial condition at $0$). Use uniqueness of solutions to finish. If $C=0$ show that the solution is $y(x)=0$.