Perturbation Solution of
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Find  an approximate solution of
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as
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The equation obviously has an analytical solution, so the purpose
here is to demonstrate the perturbation method on an equation with
a known solution, so that we can see that the method works.

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First, define the equation:

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Find the exact solutions, which comes out as a list of 2 replacment rules:

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Show how to access the element of a list (in case you do not know):

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Check that these solutions really do satisfy qe:

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Define functions x(ε)for the two exact, analytical solutions

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Now prepare to find an approximate  perturbation solution.  Note
the O[ε] symbol is powerful in Mathematica, try changing {n,0,2} to {n,0,10) and see what happens...

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Substitute the series solution for xinto the equation qe:

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LogicalExpand is really cool, and works because of the presence of the O[ε] symbol :

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The "logical and" might just as well be a list of equations that need to be true.  In fact the equation can be accessed as list, for example:

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So here are the two approximate solutions to qe:

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Compare the exact (green) and approximate (red) solutions:

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Spikey Created with Wolfram Mathematica 8.0