Finding Inner Solutions for Singular Perturbation Problems

[sara_87]

Homework Statement

\epsilon\frac{d^{2}u}{dx^{2}} +\frac{du}{dx} + e^-x = 0

0<x<1
u(0)=0
u(1)=1

Homework Equations

The Attempt at a Solution

i want to find the inner solution first
i used the substitution x=\epsilon²y

i put that in the equation:

\epsilon\frac{1}{\epsilon}^2nu'' +\frac{1}{\epsilon}^nu' +\epsilon^n y = 0
now i have to pick a value for n... how do i do that?

 

[sara_87]

epsilon is not meant to be floating that... sorry just imagine it's in line, any help would be much appreciated thanks in advance!