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I can't seem to be able to compute the inverse Laplace transform of a Laplace transform:

LaplaceTransform[x *  Sqrt[l^2 - x^2], x, s]

Out[27]= -(1/(32 π))
 Sqrt[-(1/l^2)] l^4 (l^2)^(3/2)
   s^3 MeijerG[{{-(3/2)}, {}}, {{-3, -(3/2), -1}, {}}, -(1/16)
      l^4 s^4 Sign[Im[Log[-(1/l^2)]]], 2] Sign[Im[Log[-(1/l^2)]]]

But if I try to do the reverse:

InverseLaplaceTransform[-(1/(32 π)) Sqrt[-(1/l^2)]
   l^4 (l^2)^(3/2)
   s^3 MeijerG[{{-(3/2)}, {}}, {{-3, -(3/2), -1}, {}}, -(1/16)
      l^4 s^4 Sign[Im[Log[-(1/l^2)]]], 2] Sign[
   Im[Log[-(1/l^2)]]], s, x]

Out[29]= -(1/(32 π))
 Sqrt[-(1/l^2)] l^4 (l^2)^(3/2)
   InverseLaplaceTransform[
   s^3 MeijerG[{{-(3/2)}, {}}, {{-3, -(3/2), -1}, {}}, -(1/16)
        l^4 s^4 Sign[Im[Log[-(1/l^2)]]], 2], s, x] Sign[
   Im[Log[-(1/l^2)]]]

Shouldn't this evaluate to the original? As to why I'd want to do this: I am trying to solve an equation using Laplace transforms.

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1  
Things like this actually happen far more frequently than you'd expect: operation so-and-so is easy to do, but the inverse is not too easy... – J. M.♦ Jun 27 '12 at 23:50
I agree, but is there an alternative to Laplace transforms for solving complicated equations? – highBandWidth Jun 28 '12 at 0:07
1  
@J.M. Like marriage ... – belisarius Jun 28 '12 at 1:02
2  
@belisarius ... or a lobotomy. :-p – Mr.Wizard♦ Jun 28 '12 at 7:34
1  
1. Convert it to a differential equation if possible. 2. If you cannot get a symbolic solution for the Laplace transform, then try getting a numerical solution. Write out the integral transform for a Laplace transform and try to apply it numerically to your equations using NIntegrate. – Searke Jun 28 '12 at 13:18
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