= Solution
{c}
Dropping the small highest-derivative term gives $y_o'+xy_o=0$. The <outer solution> satisfying the right endpoint condition is
$$
y_o(x)=e^{(1-x^2)/2}.
$$
It fails the left endpoint condition, so a <boundary layer> is needed at $x=0$. If $X=x/\epsilon^\gamma$, diffusion and first <derivative> balance when $\epsilon^{1-2\gamma}=\epsilon^{-\gamma}$, giving $\gamma=1$.
With $y(x)=Y(X)$, the exact <inner expansion> after multiplication by $\epsilon$ is
$$
Y_{XX}+Y_X+\epsilon^2XY=0.
$$
Its leading solution is $Y=A+B e^{-X}$. The zero boundary value gives $B=-A$, and <asymptotic matching condition> as $X\to\infty$, $x\to0$ fixes $A=e^{1/2}$. Thus
$$
Y(X)=e^{1/2}(1-e^{-X}).
$$
The common <overlap region> value is $e^{1/2}$. Add the outer and inner approximations and subtract that <overlap region>:
$$
\boxed{y_c(x)=e^{(1-x^2)/2}-e^{1/2-x/\epsilon}.}
$$
The decaying inner <exponential function> can accommodate the left condition, whereas a right-end inner coordinate would give a mode that grows into the interval. There is no right <boundary layer>. The <composite asymptotic expansion> satisfies the left condition exactly and the right one up to the exponentially small term $e^{1/2-1/\epsilon}$; the remaining approximation error is algebraic and vanishes as $\epsilon\downarrow0$.
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