Perturbation Solution of

Find an approximate solution of

as

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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${{x -> 1/2 (-ϵ - (4 + ϵ^2)^(1/2))}, {x -> 1/2 (-ϵ + (4 + ϵ^2)^(1/2))}}$

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

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${x -> 1/2 (-ϵ + (4 + ϵ^2)^(1/2))}$

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Check that these solutions really do satisfy qe:

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Define functions 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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$xs = Sum[x _ n ϵ^n, {n, 0, 2}] + O[ϵ]^3$

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Substitute the series solution for into 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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Obviously, the above equation has solutionx and .
Taking $FormBox[RowBox[{x _ 0, =, RowBox[{1, we, , solve, , for, , Cell[TextData[Cell[BoxData[x ]]]]}]}], TraditionalForm] 1$ and :