= Solution
Assume $\epsilon\downarrow0$ and fixed $A>0$. Away from the degenerate endpoint $x=0$, the reduced equation is $Ay_0'+y_0=0$. The positive drift makes the rapidly varying homogeneous mode decay as one moves away from the left boundary, so the <outer solution> is selected by the right <boundary condition>:
$$
\boxed{y_{\rm out,0}(x)=e^{(1-x)/A}.}
$$
It does not satisfy the left value: its limit there is $L=e^{1/A}$.
If the layer width is $\delta$, diffusion is $O(\epsilon/\delta^2)$, while $Axy'$ is $O(A)$ because the drift vanishes linearly. Their <dominant balance> gives the <square-root boundary layer at a vanishing drift>
$$
\boxed{x=O(\sqrt{\epsilon/A}),\qquad\xi=x\sqrt{A/\epsilon}.}
$$
Writing $y=Y(\xi)$ gives $Y''+\xi Y'+[\sqrt\epsilon/A^{3/2}]\xi Y=0$. The first nonzero <inner expansion> satisfies $Y_0''+\xi Y_0'=0$, hence $Y_0'=Be^{-\xi^2/2}$. Imposing $Y_0(0)=1$ and matching $Y_0(\infty)=L$ gives
$$
\boxed{Y_0(\xi)=1+(e^{1/A}-1)\operatorname{erf}(\xi/\sqrt2).}
$$
The <overlap region> is $\sqrt{\epsilon/A}\ll x\ll1$ for fixed $A$. The <additive composite expansion>
$$
y_{\rm comp,0}=e^{(1-x)/A}+(1-e^{1/A})\operatorname{erfc}\!\left(x\sqrt{\frac A{2\epsilon}}\right)
$$
combines the two leading approximations and satisfies the left value exactly and the right value up to an exponentially small term. Only the first nonzero terms were requested; higher matching terms need not be added.
For fixed $A<0$, the decaying fast mode is at the other endpoint. The <outer solution> is now fixed by $y(0)=1$, giving $e^{-x/A}$. At $x=1$ the drift does not vanish, so the right layer has width $O(\epsilon/|A|)$, with leading transient proportional to $e^{-|A|(1-x)/\epsilon}$. In particular its leading inner profile is $e^{-1/A}+(1-e^{-1/A})e^{-\eta}$, where $\eta=|A|(1-x)/\epsilon$. The vanishing drift at the left endpoint does not require an order-one mismatch layer for these data.
For $A=0$, the reduced equation is $xy_0=0$, which cannot supply the nonzero endpoint values. The equation becomes an <Airy ordinary differential equation>: $x=\epsilon^{1/3}\xi$ gives $Y''+\xi Y=0$. The exact solutions are combinations of $\operatorname{Ai}(-x/\epsilon^{1/3})$ and $\operatorname{Bi}(-x/\epsilon^{1/3})$. For $x>0$ they oscillate, with local wavelength of order $\sqrt{\epsilon/x}$, rather than forming a decaying exponential layer matched to a smooth nonzero outer profile. The $O(\epsilon^{1/3})$ region resolves the turning endpoint, but oscillations extend throughout the interval. Boundary-value resonances can occur at isolated parameter values. Thus setting $A=0$ is not a continuous substitution into either fixed-sign layer approximation.
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