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Finding general solution to this pde

  1. Jun 10, 2012 #1
    Hi, my equation is;

    [tex]
    \frac{\partial}{\partial t}U(x,y,t) = 2g \left( x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x} \right) U(x,y,t)
    [/tex]

    I want to find the general solution to this but I don't know how to find it?

    Any help would be great...thanks :D
     
  2. jcsd
  3. Jun 10, 2012 #2

    micromass

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    What class is this for, a PDE class???

    What have you tried already?
     
  4. Jun 11, 2012 #3
    Yer its PDE class.

    I've tried using Charactoristics, so,

    [tex]
    \frac{\partial x(t)}{\partial s} = 2gy(t)
    [/tex]

    and

    [tex]
    \frac{\partial y(t)}{\partial s} = -2gx(t)
    [/tex]

    With s=t. I am not sure what to do now with regards to forming a general solution, would it be something of the form,

    [tex]
    U(x,y,t) = f (x_0 +2gy(t)t, y_0 -2gx(t)t, t)
    [/tex]

    I'm not sure how to proceed.

    Thanks.
     
  5. Jun 11, 2012 #4

    fzero

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    Writing the original equation in polar coordinates should be illuminating. There's a further linear change of variables that will put the equation into a form where the general solution should be obvious.
     
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