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

The characteristic equations are given by:
$$ dx/x=dy/y=dz/1=du/u.$$
$dx/x=dy/y \implies \ln(x)=\ln(y)+\ln(c_1) \implies x/y = c_1$
$dy/y =dz/1 \implies \ln(y)=z+c_2 \implies c_2 = \ln(y)-z$
$dz/1=du/u \implies z=ln(u)-\ln(c_3) \implies u = c_3e^{z}$
Now use that $c_3=F(c_1,c_2)$, hence:
$u=F(c_1,c_2)e^{z}=F(x/y,\ln(y)-z)e^{z}$
