PDEMOC11

$(x+u)u_x + (y+u)u_y = 0\,$

The characteristics are $\frac{dx}{dt}=x+z, \frac{dy}{dt}=y+z, \frac{dz}{dt}=0\,$ .

$\ln(x+z)=t+c_1, \ln(y+z)=t+c_2, z(t)=c_3\,$

$x+z=c_4e^t, y+z=c_5e^t\,$

$y+z=c_6(x+z)\,$

$\frac{y+z}{x+z} = c_6\,$

So we have two functions that are LI and constant on the surface. Set one equal to an arbitrary function $\phi\,$ of the other:

$z=u(x,y,z)=\phi\left(\frac{y+z}{x+z}\right)\,$