Find general solution
\begin{equation}
4 u_x -3u_y=0;
\label{eq-HA1.1}
\end{equation}
Solve IVP problem $u|_{x=0}=\sin (y)$ for equation (\ref{eq-HA1.1}) in $\mathbb{R}^2$;
Consider equation (\ref{eq-HA1.1}) in $\{x>0, y>0\}$ with the initial condition $u|_{x=0}=y$ ($y>0$); where this solution defined? Is it defined everywhere in $\{x>0, y>0\}$ or do we need to impose condition at $y=0$?
In the latter case impose condition $u|_{y=0}=x$ ($x>0$) and solve this IVBP;
Consider equation (\ref{eq-HA1.1}) in $\{x<0, y>0\}$ with the initial condition $u|_{x=0}=y$ ($y>0$); where this solution defined? Is it defined everywhere in $\{x<0, y>0\}$ or do we need to impose condition at $y=0$? In the latter case impose condition $u|_{y=0}=x$ ($x<0$) and solve this IVBP.
Find the general solution of
\begin{equation}
xu_x+4 yu_y=0
\label{eq-HA1.2}
\end{equation}
in $ \{(x,y)\ne (0,0)\}$; when this solution is continuous at $(0,0)$?
Find the general solution of
\begin{equation}
xu_x-4yu_y=0
\label{eq-HA1.3}
\end{equation}
in $ \{(x,y)\ne (0,0)\}$; when this solution is continuous at $(0,0)$?
Explain the difference between (\ref{eq-HA1.2}) and (\ref{eq-HA1.3}).
Find the general solution of
\begin{equation}
u_{tt}-9u_{xx}=0;
\label{eq-HA1.7}
\end{equation}
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});
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.