Solve PDE: $xu_x + yu_y + u_z = u , u(x,y,0)=h(x,y)$ Here is problem: solve PDE, quasilinear, problem
$xu_x + yu_y + u_z = u , u(x,y,0)=h(x,y)$.
Here what I did: Given $\Gamma: <x=s, y=s, z =0, u=h(s)>$
$dx/dt =x$, $dy/dt = y$, $dz/dt=1$ and $du/dt = u$
$x=se^t$, $y=se^t$, $z=t$ , $u=h(s)e^t$. Now I am stuck. I do not know to get s and solution. Please help. Thanks

You have the wrong initial curve. It is $\Gamma:(s_1,s_2)\mapsto (s_1,s_2,0)$ and $u|_\Gamma=h(s_1,s_2)$. 
You have computed the ODEs correctly (up to the initial condition), we have $x=s_1e^t$, $y=s_2e^{t}$, $z=t$ and $u=h(s_1,s_2)e^t$. As you noticed, you have to solve for $s_1$ and $s_2$. We have \begin{align} s_1=xe^{-t}=xe^{-z}, \\ s_2=ye^{-t}=ye^{-z}.\end{align}
Thus plugging this into $u$ yields
$$u(x,y)=h(xe^{-z},ye^{-z})e^z.$$
