= Solution
{c}
For the left inner coordinate $X=x/\delta$, the equation becomes $Y_{XXX}-Y_X+\delta^2XY=0$. At leading order the solution that matches a bounded outer expansion is $Y_0=A+De^{-X}$; the growing exponential is excluded. The boundary value gives $A+D=1$. Since $y'(0)=1$ is finite, the possible $O(\delta^{-1})$ <derivative> must vanish, so $D=0$ and $A=1$.
At the next order, $Y_{1/2}=B+Ee^{-X}$. Its boundary value is zero, giving $E=-B$, while its physical <derivative> at zero is $-E=1$. Hence $B=1$, and the left inner expansion is $1+\delta(1-e^{-X})+O(\delta^2)$.
At the right endpoint put $\xi=(1-x)/\delta$. The equation is $Y_{\xi\xi\xi}-Y_\xi=\delta(1-\delta\xi)Y$. Matching the outer expansion at one and imposing $Y(0)=1$ give
$$
Y_0=\sqrt e+(1-\sqrt e)e^{-\xi}.
$$
At the next order, $Y_{1/2}^{(3)}-Y_{1/2}'=Y_0$. The solution with no growing exponential, the correct matching polynomial, and $Y_{1/2}(0)=0$ is
$$
Y_{1/2}=\sqrt e(1-\xi)+
\left[\frac{1-\sqrt e}{2}\xi-\sqrt e\right]e^{-\xi}.
$$
Indeed the operator $D^3-D$ sends $-\sqrt e\,\xi$ to $\sqrt e$ and sends $\xi e^{-\xi}$ to $2e^{-\xi}$.
Add the two decaying inner corrections to the <outer solution>, subtracting their common matching parts. Since $\delta\xi=1-x$, the <composite asymptotic expansion> is
$$
\boxed{\begin{aligned}
y_c(x)={}&e^{x^2/2}(1+\delta)-\delta e^{-x/\delta}\\
&+e^{-(1-x)/\delta}\left[(\sqrt e-1)\left(\frac x2-\frac32\right)-\delta\sqrt e\right].
\end{aligned}}
$$
It reproduces all three boundary conditions to the retained algebraic orders, with exponentially small interactions between endpoints. The right endpoint has an $O(1)$ <boundary layer> in $y$. The left correction has only <amplitude> $O(\sqrt\epsilon)$ in $y$, but its <derivative> is $O(1)$ and is necessary for the <derivative> <boundary condition>. Thus the left layer is a <weak derivative boundary layer>; if “genuine” refers specifically to an order-one change in the function itself, only the right layer is genuine in that sense. Both nonzero corrections belong in the expansion through $O(\sqrt\epsilon)$.
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