[SOLVED] PDE using method of characteristic strips

Hi guys. I'd like to solve the following:

Quote:

$u=xu_x+yu_y+\frac{1}{2}((u_x)^2+(u_y)^2)$ ,

$u(x,0)=\frac{1}{2}(1-x^2)$ .

This corresponds to the problem $F=xp+yq+\frac{1}{2}(p^2+q^2)-z=0$ .

Then the characteristic equations are given by

$\frac{dx}{dt}=x+p$ , $\frac{dy}{dt}=y+q$ , $\frac{dz}{dt}=p(x+p)+q(y+q)$ , $\frac{dp}{dt}=0$ , $\frac{dq}{dt}=0$ .

And it's easy to find the curves (there are two) induced by the initial condition:

$\Gamma=(x(s,0),y(s,0),z(s,0),p(s,0),q(s,0))=(s,0, \frac{1}{2}(1-s^2),-s,\pm 1)$

Solving for $z$ we have

$z=-xs\pm y+\frac{1}{2}(s^2+1)$

(where the signs $\pm$ correspond to the signs in $\Gamma$ ).

But I don't know how to put this in closed form (i.e. in the form $z=z(x,y)$ ). It may just be a simple elementary calculus/algebra block on my part. Any help would be much appreciated!