Consider the equation of the form $\epsilon y^{\prime \prime}+a y^{\prime}=0$ on $x\in[0,1]$ with $a\in\mathbb{R}$, $0<\epsilon\ll1$,$y(0) =\alpha$, and $y(1)=\beta$. Show that if $a >0$ then the boundary layer is near $x= 0$ with the method of matched asymptotic expansions.
To find the inner solution, Wikipedia defines a new variable $\tau=x/\epsilon$, but why not $x/\epsilon^{2}$ or $x/\epsilon^{3}$? I already know it depends on the thickness of the boundary layer, but for my problem above, how can I determine the thickness? (How can I define a new variable to find the inner solution? Is $\tau=x/\epsilon$ okay?)
I got another question just now, i.e. how can I prove the boundary layer is near $x=0$? When I have the outer solution, there exists two boundary condition but I don't know which one should be used. How can I determine which one is the "outer"?
You are right that you can not necessarily just "see" what the thickness $\tau$, the exponent in $\delta=ϵ^τ$, of the boundary layer is. You would then need to make the general zooming coordinate transform $x=x_0+δX$ and consider the ODE for $Y(X)=y(x)$ which in the current case gives $$ ϵY''(X)+aδY'(X)=0. $$ The enjoining reasoning is that $Y$, having the same values as $y$, is bounded and that this coordinate transform renders also the derivatives of $Y$ bounded relative to $ϵ\to 0$. For that being the case, the terms in the equation have to balance, $ϵ$ and $δ$ need to be on the same scale. This is easiest achieved by just setting $δ=ϵ$. Thus $Y(X)=Ce^{-aX}+D$, $C\ne 0$ (As $C=0$ implies that no boundary layer exists at this point).
Now one needs to ensure that the boundedness assumption is actually satisfied for this inner solution. If $x_0$ is an inner point, this requires $\lim_{X\to\pm \infty}Y(X)$ being finite/existing, which is not the case here. Thus $x_0$ has to be one of the boundary points, and only the limit towards the inside the interval has to be finite, which is feasible.