Solution
= Solution
Choose $F(s)=G(s)=\tfrac12\sin s$. The two travelling profiles then add by the <sine> addition identity:
$$
\boxed{y(x,t)=\frac12\sin(2x+5t)+\frac12\sin(2x-5t)
=\sin2x\cos5t.}
$$
This is a <standing wave>. At $x=0$ and $x=\pi$ it vanishes for all $t$. At $t=0$ its displacement is $\sin2x$, and its time <derivative> $-5\sin2x\sin5t$ is zero. Also $y_{tt}=-25y$ and $y_{xx}=-4y$, directly checking the original equation.