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Partial Differential Equation -MOC

  1. Dec 2, 2011 #1
    1. The problem statement, all variables and given/known data
    The PDE: [itex]{\partial \rho \over \partial t} + \rho {\partial \rho \over \partial x} =-x\rho[/itex]
    and [itex]\rho(x,0) = f(x)[/itex]
    Determine a parametric representation of the solution.

    2. Relevant equations
    [itex]{dx\over dt} = \rho[/itex]

    [itex]{d\rho \over dt} = -x\rho[/itex]

    3. The attempt at a solution
    Using method of characteristics I get

    [itex]\rho = \rho(x_0,0)\exp(-xt) = f(x_0)\exp(-xt)[/itex]

    [itex]x=-f(x_0)\exp(-xt)/x + c[/itex]

    And since at t=0 x=x_0
    [itex]x=-{f(x_0)\over x}(1-\exp(-xt))+x_0[/itex]

    This is my solution, but if i plug in f.x f(x) = x and substitute ρ into the PDE, i get rubish.

    If f(x)=x, then

    [itex]x_0 = {x\over (1-\exp(-xt))/x +1}[/itex]

    [itex]\rho = x_0 \exp(-xt)[/itex]

    And this substituting this into
    [itex]{\partial \rho \over \partial t} + \rho {\partial \rho \over \partial x} +x\rho[/itex]

    Gives a non-zero term.

    Any idea what I'm doing wrong?
     
  2. jcsd
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