Separation of variables to derive the solution $u(x, t)$ How do I solve this question, what are the steps ?
Use the method of separation of variables to derive the solution $u(x, t)$ to the equation for a vibrating string
$$\frac{\partial ^2u}{\partial \:t^2}=9\:\frac{\partial ^2u}{\partial x^2} \hspace{4em} (0 < x < 4, t > 0)$$
with fixed endpoints $u(0, t) = u(4, t) = 0$. The initial conditions are given by
 $u(x, 0) =\frac{1}{2}\left(\frac{\pi \:x}{4}\right)$
$$\frac{\partial u}{\partial t}\:\left(x,\:0\right)\:=\:−\:\sin\left(\frac{\pi x}{2}\right)\:\hspace{4em} (0 ≤ x ≤ 4)$$

I advise you to go through the procedure outlined here (first hit on Google when you search for 'separation of variables'), which shows the method for the equation
\begin{equation}
 \frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}.
\end{equation}
Adopting it to your problem shouldn't be too hard.
