## Existence and Uniqueness Theorem

Let $\phi_{1}$ and $\phi_{2}$ be two solutions on $\mathbb{R}$ of the differential equation
$y^{''} + sin^{2}(x)y = 0$,
such that $\phi_{1}(0) = 1$, $\phi^{'}_{1}(0) = 0$ and $\phi_{2}(0) = 0$, $\phi^{'}_{2}(0) = 1$. Let $\phi(x)$ be the function defined by
$\phi(x) = \phi_{2}(\pi)\phi_{1}(x) + (1 - \phi_{1}(\pi))\phi_{2}(x)$.
(a) Show that $\phi$ is also a solution of $y^{''} + sin^{2}(x)y = 0$.
(b) Suppose $\phi_{1}(\pi) + \phi^{'}_{2}(\pi) = 2$. Show that $\phi(x) = \phi(x + \pi)$ for all $x \in \mathbb{R}$.

I've managed to do part (a) and the part $\phi(0) = \phi(\pi)$. However, I have difficulty in proving that $\phi^{'}(0)=\phi^{'}(\pi)$ using the condition stated in part (b) in order to show that $\phi(x) = \phi(x + \pi)$ using the existence and uniqueness theorem. Could anyone help with this?