## Boundary Value problem and ODE

Hi, I'm not sure if this is on the right thread but here goes. It's a perturbation type problem.

Consider the boundry value problem

$$\epsilon y'' + y' + y = 0$$
Show that if $$\epsilon = 0$$ the first order constant coefficient equation has
the solution
$$y_{outer} (x) = e^{1-x}$$
I have done this fine.
Find a suitable rescaling $$X = x/\delta$$ so that the highest derivative is important and balances another term and find the solutions $y_{inner}(X)$ of
this equation, containing one free parameter, satisfying the boundary
condition at $$x = X = 0$$

So I am at the rescaling part and solved the differential equation (after neglecting the δy part of the full equation)
$$\frac{d^2y}{dX^2} + \frac{dy}{dX} = 0$$
yielding $$y = Ae^{-X} + B$$

imposing the boundry condition $$x = X = 0$$

gives $$A = B$$

so is $$y_{inner} = Ae^{-X}$$ ??
I think I covered that the highest derivative is important (Although again I was unsure about the wording)

When I continue on I think I either have this part wrong or the matching is wrong as I am not getting the right answer. I am supposed to be doing intermediate scaling. If anyone has any comments about this that would also be much appreciated.