= Solution
{c}
For the positive coefficient of $y''$, setting $\epsilon=0$ gives the reduced equation $y_0'+xy_0=0$, with <general solution> $y_0=Ce^{-x^2/2}$. A fast decaying <homogeneous solution> of $\epsilon y''+y'=0$ is $e^{-x/\epsilon}$, placing the <boundary layer> at the left endpoint. Thus use the right <boundary condition> for the <outer solution>: $y_0(1)=0$ implies
$$
\boxed{y_{\mathrm{out},0}(x)=0}.
$$
The <dominant balance> $\epsilon/\delta^2\sim1/\delta$ gives width $\delta=\epsilon$. With the <inner variable> $X=x/\epsilon$ and $Y(X)=y(\epsilon X)$, the exact rescaled equation is
$$
Y_{XX}+Y_X+\epsilon^2XY=0.
$$
At leading order, $Y_{0,XX}+Y_{0,X}=0$, so $Y_0=A+Be^{-X}$. The <asymptotic matching condition> $Y_0(\infty)=y_{\mathrm{out},0}(0)=0$ sets $A=0$, and $Y_0(0)=1$ sets $B=1$. Therefore
$$
\boxed{Y_0(X)=e^{-X},\qquad y_{\mathrm{comp},0}(x)=e^{-x/\epsilon}}.
$$
The <additive composite expansion> has zero overlap to subtract. Its graph starts at one, decreases steeply over $x=O(\epsilon)$, and then stays close to zero; its initial slope is $-1/\epsilon$. The right endpoint discrepancy $e^{-1/\epsilon}$ is exponentially small. The left panel of the figure below sketches this profile. A right-end <inner expansion> would instead contain a growing $e^X$ term and could not repair the left endpoint mismatch.
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