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The ONLY solution of PDE: f_t g_r = f_r g_t ?

  1. Jul 5, 2008 #1
    The ONLY solution of PDE: f_t g_r = f_r g_t ???

    I have the following PDE:

    [tex]\frac{\partial f(r,t)}{\partial t} \, \, \frac{\partial g(r,t)}{\partial r} = \frac{\partial f(r,t)}{\partial r} \, \, \frac{\partial g(r,t)}{\partial t}[/tex]

    By a simple check, I know a solution is f = h(g), where h() is arbitrary function. The Maple PDE solver returns exactly that.

    How can I prove, f=h(g) is the ONLY type of solution of that PDE?
     
    Last edited: Jul 5, 2008
  2. jcsd
  3. Jul 8, 2008 #2
    Re: The ONLY solution of PDE: f_t g_r = f_r g_t ???

    Your statement is equivalent to

    [tex] \frac{\partial (f,g)}{\partial (t,\tau)} = 0 [/tex]
     
  4. Jul 8, 2008 #3
    Re: The ONLY solution of PDE: f_t g_r = f_r g_t ???

    Thanks!
     
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