I need all solutions of $(\partial\_x u)^2+(\partial\_y u)^2=1$ for the function $u(x,y)$. Of course I know simple solutions like $u=ax \pm \sqrt{1-a^2}y + c$, or $u=\sqrt{x^2+y^2}+c$; but what's the general solution?


More generally, I'd like to know how to tackle a PDE of the form $|\nabla u|^2=f^2(u)$ where $f$ is some given function. Again a simple solution is to assume that $u$ is a function of $r=\sqrt{x^2+y^2}$ and integrate the equation. But what's the most general solution?



Solve |\nabla u|^2=1
$|\nabla u|^2=f(u(r))$ is a special case of the eikonal equation. 
You could advise any good book on pdes. 
Also you will need to sharpen your knowledge on Hamilton-Jacobi methods.


ps. Oops, beaten to it.


