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Taylor series at a point for which the function isn't defined (perturbation)

  1. Oct 14, 2011 #1
    1. The problem statement, all variables and given/known data

    This problem arises from the following ODE:

    [tex]\epsilon y'' + y' + y = 0, y(0) = \alpha, y(1) = \beta[/tex]

    where [itex]0 < x < 1, 0 < \epsilon \ll 1[/itex]

    Find the exact solution and expand it in a Taylor series for small [itex]\epsilon[/itex]

    2. Relevant equations

    I guess knowing the Taylor series formula would be helpful

    3. The attempt at a solution

    Using ye olde constant coefficient method, I get that the solution (in non-expanded form) is:

    [tex]y(x) = c_1\cosh(rx) + c_2\sinh(rx)[/tex]

    where

    [tex]r = \frac{-1 \pm \sqrt{1-4\epsilon}}{2\epsilon}[/tex]

    (this is real since [itex]\epsilon[/itex] is so small)

    Applying the boundary conditions gives that

    [tex]y(x) = \alpha\cosh(rx) + \frac{\beta-\alpha\cosh(r)}{\sinh(r)}\sinh(rx)[/tex]

    Now the goal is to do a Taylor series not for x, but for epsilon (remember that r is a function of epsilon). Trouble is that this diverges near [itex]\epsilon = 0[/itex], so I don't see how to do it. I tried putting it into Mathematica and got a very crazy answer. I'm not sure it's right, and I'm less sure how to get it.

    Thanks! :)
     
  2. jcsd
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