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

1. Oct 14, 2011

### tjackson3

1. The problem statement, all variables and given/known data

This problem arises from the following ODE:

$$\epsilon y'' + y' + y = 0, y(0) = \alpha, y(1) = \beta$$

where $0 < x < 1, 0 < \epsilon \ll 1$

Find the exact solution and expand it in a Taylor series for small $\epsilon$

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:

$$y(x) = c_1\cosh(rx) + c_2\sinh(rx)$$

where

$$r = \frac{-1 \pm \sqrt{1-4\epsilon}}{2\epsilon}$$

(this is real since $\epsilon$ is so small)

Applying the boundary conditions gives that

$$y(x) = \alpha\cosh(rx) + \frac{\beta-\alpha\cosh(r)}{\sinh(r)}\sinh(rx)$$

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 $\epsilon = 0$, 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! :)