Solving fully nonlinear 1st order.

Question

The problem is the following:

Consider the PDE $c^2(u_x{^2}+u_y{^2}+u_z{^2})=1,c$ is a constant, which arises in geometrical optic. Suppose that on the sphere of radius $r$,$u=u^0=\alpha$ is a constant. Determine u.

What's method do i solve ?

Answer 1

We search solution $u=f(x^2+y^2+z^2)$. Then we get two solutions: $$u=\frac{\sqrt{x^2+y^2+z^2}}{c}-\frac{r}{c}+\alpha$$ and $$u=-\frac{\sqrt{x^2+y^2+z^2}}{c}+\frac{r}{c}+\alpha$$