Solutions to be posted as a "Reply" only after January 22, 21:00

**a.** Find the general solution of

\begin{equation}

u_{tt}-9u_{xx}=0;

\label{eq-HA1.7}

\end{equation}

**b.** Solve IVP

\begin{equation}

u|_{t=0}=\sin(x),\quad u_t|_{t=0}=\cos(x)

\label{eq-HA1.8}

\end{equation}

for (\ref{eq-HA1.7});

**c.** Consider (\ref{eq-HA1.7}) in $\{t>0, \, 3t> x > -3t\}$ and find a solution to it, satisfying Goursat problem

\begin{equation}

u|_{x=3t}=t,\quad u|_{x=-3t}=6t.

\label{eq-HA1.9}

\end{equation}

**Remark.**

Goursat problem for wave equation $u_{tt}-c^2u_{xx}=0$ in ${t> 0, -ct<x<ct}$ is $u|_{x=ct, t>0}=\phi(t)$, $u|_{x=-ct, t>0}=\psi(t)$ and one often assumes that compatibility condition $\phi(0)=\psi(0)$ is fulfilled. It is very important that $x=\pm ct$ are characteristics.