## Spring 2016, problem 16

Let $f : \mathbf{R} \rightarrow \mathbf{R}$ be a twice differentiable, $2 \pi$-periodic, even function. Prove that if $$f''(x)+f(x)=\frac{1}{f(x+ \frac{3\pi}{2} )}$$ holds for every $x$, then $f$ is actually $\frac{\pi}{2}$-periodic.