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Solving the Poisson equation (second order ODE)

Posted on 2009-12-17
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Last Modified: 2012-05-08
I need to solve Poisson's equation in spherical coordinates (which is modelling something related to plasmas). The equation is:

   1/r^2 d/dr(r^2 * dp(r)/dr) = 2p(r)/k^2

where k is a known constant. So the first thing seems to be to take the r^2 to the other side, and apply the product rule on the left, to get the second order ODE:

   2r p' + r^2 p'' = 2r^2 p/k^2

or

   p'' + (2/r)p' - (2/k^2)p = 0.                                                 (*)

How do I solve this? According to Wolfram, this has the general solution:

   p(r) = c1*exp(-sqrt(2)*r/k)/r + c2*exp(sqrt(2)*r/k)/(2*sqrt(2)*r/k)                           *EDITED*

which indeed leads to the correct solution when I apply some known conditions, but how can I derive this solution?

Thanks very much
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Question by:InteractiveMind
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by:aburr
ID: 26077625
"how can I derive this solution?"
Differentiate it?
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by:InteractiveMind
ID: 26083143
Differentiate (*)? What then?
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by:aburr
ID: 26085492
You said you thought you had the general solution
" p(r) = c1*exp(-sqrt(2)*r/k)/r + c2*exp(sqrt(2)*r/k)/(2*sqrt(2)*r/k)"
so if you differentiate it and get
"  p'' + (2/r)p' - (2/k^2)p = 0.       "
you thought correctly
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by:InteractiveMind
ID: 26086538
Sorry, no, I wanted to know if the general solution can be derived. I always hate it when we have to pull a solution out of the hat and justify it on the basis that it works (Ansatz).
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aburr earned 2000 total points
ID: 26087860
"I always hate it when we have to pull a solution out of the hat and justify it on the basis that it works (Ansatz)."
I do too but it is often the easiest way. Integration is the inverse of differentiation and inverse operations are always harder. Consider long division, the inverse of multiplication. It is just an organized trial and error procedure. The same with integration. Poisson's equation can just be looked at as the result of an organized trial and error.
In general physics SHM equation is usually just plopped down (and checked by differentiation).
The whole practice of integration is the result of trail and errors. 1st order, 2nd order, homogeneous, non-homogeneous, separable, etc)
You may be able to get somebody to help you further (but not me)'

Note which might lead to something
If the variables cannot be separated, the method of integral transforms might help.
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