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PDE: The Eikonal Equation method of characterstics, etc.

  1. Feb 4, 2010 #1
    1. The problem statement, all variables and given/known data
    I need to solve the Eikonal Equation [tex]c^2(u_x^2 + u_y^2) = 1[/tex]

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

    Oh. And the solution is given as [tex]\ln{\frac{\sqrt{x^2 + y^2} + y}{x}}[/tex]
    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.
  2. jcsd
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