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Simple Partial Differential Equation

  1. Oct 12, 2014 #1
    1. The problem statement, all variables and given/known data
    This is actually an electromagnetism problem but all the physics is done, I just don't remember how to solve the PDE:
    [itex]\frac{d^2V}{dr^2}=-\frac{2}{r}\frac{dV}{dr}[/itex]
    The d's should be del's, just don't know how to do that...

    2. Relevant equations

    Not sure.

    3. The attempt at a solution

    Don't know where to start.
     
  2. jcsd
  3. Oct 12, 2014 #2

    Dick

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    There are no other independent variables in the problem besides r. Assume V can be written as a product of functions of all the independent variables. I.e. it's separable. V(r)=u(r)*v(theta)*w(phi). The treat it as an ODE. Do you remember those? I would try putting dV/dr=u(r) and separating variables.
     
    Last edited: Oct 12, 2014
  4. Oct 12, 2014 #3
    Ahh I think that's just what I needed. So

    [itex]V_{rr}=-k^2u(r)[/itex] where k=sqrt{2/r}
    [itex]u(r)=Ae^{ikr}+Be^{-ikr}[/itex]
    [itex]V(r)=\frac{-iA}{k}e^{ikr}+\frac{iB}{k}e^{-ikr}[/itex]
    then sub in for k

    Is that correct?
     
  5. Oct 12, 2014 #4

    Dick

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    Not really. If u(r)=dV/dr then the equation becomes du/dr=(-2/r)u. Doesn't it? Solve that.
     
  6. Oct 12, 2014 #5
    Ahh, I'm an idiot. [itex]u=r^{-2}[/itex]

    Thanks Dick!
     
  7. Oct 12, 2014 #6

    Dick

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    You're welcome but don't forget the arbitrary constant when you integrate.
     
  8. Oct 12, 2014 #7

    Ray Vickson

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    Note that
    [tex] \frac{d}{dr}\left( r^k \frac{dF(r)}{dr} \right) = r^k \frac{d^2F(r)}{dr^2}+ k r^{k-1} \frac{dF(r)}{dr}[/tex]
    Can you see how that fact helps?
     
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