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.