Consider the equation
$(x+1-\epsilon)\frac{dy}{dx}+(1-\frac{1}{4}\epsilon^2y)y=2(1-\epsilon x)$
with $y(1)=1$.
I am interested in finding an asymptotic expansion for the inner solution so I put $x= \epsilon^{\alpha} X$. My question here is how I do manage to determine the value of $\alpha$? Or equivalently, what is the thickness of the inner layer?