# Math Help - 2nd order ODE (Help)

1. ## 2nd order ODE (Help)

Let $\phi (x)$ be a solution of the second order linear differential equation
$y^{''}+(sin x)^{2}y^{'}+y=0$
defined on $\mathbb{R}$ satisfying $\phi (0)=\phi (\pi)$ and $\phi^{'} (0)=\phi^{'} (\pi)$. Prove that $\phi (x+\pi)=\phi (x)$ for all $x \in \mathbb{R}$.

How do I go about doing it?

2. Let $\psi(x)=\phi(x+\pi), \ x\in \mathbb{R}$.
Then $\psi(0)=\phi(\pi)=\phi(0), \ \psi'(0)=\phi'(\pi)=\phi'(0)$.
By uniqueness, $\psi\equiv\phi$.