# PDE: The Eikonal Equation method of characterstics, etc.

1. Feb 4, 2010

### bndnchrs

1. The problem statement, all variables and given/known data
I need to solve the Eikonal Equation $$c^2(u_x^2 + u_y^2) = 1$$

Initial condition u(x,0) = 0 C(x,y) = |x|, but x>0 to essentially C = x

Oh. And the solution is given as $$\ln{\frac{\sqrt{x^2 + y^2} + y}{x}}$$
2. Relevant equations
None other than the usual method of characteristics stuff

3. The attempt at a solution

I can go through the method of characteristics and I get stuck with solving for X(s,t) and Y(s,t) at:

dX/dt = P*X^2
dY/dt = X^2*1/s
dP/dt = -1/X

s is a parameter here, not a variable. I'm really stuck, and need some fresh insights on this one, I've been working it for too long that I'm missing something critical.