A Regular Perturbation Expansion
for

Solution of with
y(0)=1.
We seek a solution of form

Logan 1.4 on page 50 (1st edition).

Here is the proposed series solution:

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Here is the equation with the proposed perturbation solution:

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With the O[ε] symbol, the LogicalExpand does a powerful step:

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Here is another way to look at it:

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Here is the ODE for :

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...and it's solution:

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Here is the ODE for , making use of the known solution for :

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Here is the solution for :

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Now convert these replacement rules into approximate solutions, of increasing O[ε],
that we can plot:

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Test these functions:

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Lucky for us, we can find the exact solution:

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and here it is as a convenient function:

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Let's test it:

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Here we plot the exact solution (green), the (red), (blue),  and (black).

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Created with Wolfram Mathematica 8.0