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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?

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    Basically, I know that I need to rewrite the ODE in terms of X and then do some kind of dominant balance but I get the equation $(\epsilon^{\alpha}X+1-\epsilon) \epsilon^{-\alpha}\frac{dY}{dX} + (1-\frac{1}{4}\epsilon^2 Y)Y=2(1-\epsilon \epsilon^{\alpha} X)$ and it doesn't look right.2011-12-24

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