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Method of characteristics

  1. Jul 11, 2011 #1
    1. The problem statement, all variables and given/known data

    x([itex]\partial u / \partial x [/itex]) + y([itex]\partial u / \partial y [/itex]) = -[itex]x^2u^3[/itex]where u(x,1) = x for -[itex]\infty[/itex] < x < [itex]\infty[/itex]

    2. Relevant equations

    3. The attempt at a solution

    dy/dx = y/x

    = ln(y)=ln(x)+k k=constant of integration
    =[tex] y = x + e^K[/tex]

    along this characteristic
    [tex]du/dx = -(x^2u^3)/x[/tex]

    = [tex]-xu^3[/tex]

    = [tex]1/(2u^2) = ln(x) + F(K) [/tex]

    not sure where to go from here...

    should i simplify more for u and swap in k=y-x then use the conditions?
  2. jcsd
  3. Jul 11, 2011 #2
    You're using the wrong method here, it seems. The method of characteristics you've set up is tailored to the case where the divergence of the function u is zero. Here it is not. Try something like a change of variables, to eliminate (say) y from your equation and reduce it to one variable.
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