2nd order differential equation

[sara_87]

Homework Statement

$$\epsilon$$y'' - y' = 0
y(0)=0, y(1)= 1

The Attempt at a Solution

this s a perturbation problem, but I'm not sure how to go about doing it
if we let epsilon = 0 we'll go from 2nd order to first order so it's a boundary layer problem... i think
so if that's the case then i have to find the outer and inner layer
to find the outer layer i let epslon =0 and i get y' = 0

then to find the inner layer i let x=$$\epsilon$$ⁿX

then after a few steps i got to this:
1/$$\epsilon$$ d²y/dX² - 1/$$\epsilon$$dy/dX

and then solve it normally then find the general solution etc.

and also I'm letting y = y0+ epsilony1 + epsilony2...
and y' = ...
and y''= ...

and subbing them in and solving it too... i can do that, even though there's and epsilon?
and then I'm supposed to compare these together?

 

[sara_87]

i didn't mean for the epsilon to be floating like that
the epsilon is meant to be firmly on the ground... hope this helps

 

[hunt_mat]

You have two sets of solutions, one outer solution and one inner solution.

To find the inner solution, set $$x=\delta (\varepsilon )X$$ and look for the dominant balance.