= Solution
{c}
The common value in the <overlap region> is $e$. The <additive composite expansion> therefore gives
$$
\boxed{y_{\mathrm{comp},0}(x)=y_{\mathrm{out},0}(x)+y_{\mathrm{in},0}(x)-e
=e^{x-1}-e\,e^{-2(2-x)/\epsilon}.}
$$
This leading approximation has $y_{\mathrm{comp},0}(2)=0$ and $y_{\mathrm{comp},0}(1)=1-e^{1-2/\epsilon}$; the left endpoint error is exponentially small. It rises from near one following the <outer solution> $e^{x-1}$, approaches a value close to $e$, then drops rapidly to zero in the right <boundary layer>. Its first <derivative> is
$$
y_{\mathrm{comp},0}'=e^{x-1}-\frac{2e}{\epsilon}e^{-2(2-x)/\epsilon},
\qquad y_{\mathrm{comp},0}'(2)=e\left(1-\frac2\epsilon\right).
$$
For small $\epsilon$ the largest value lies slightly inside the interval. For this <additive composite expansion>, the first <derivative> vanishes at
$$
2-x_{\max}=\frac{\epsilon\log(2/\epsilon)}{2-\epsilon}
\sim\frac\epsilon2\log\frac2\epsilon.
$$
This locates the rounded top of the sketch; the rapid final drop has the $O(\epsilon)$ scale found above. The <additive composite expansion> is a leading approximation, so its exponentially small endpoint error does not imply an exponentially small error throughout the interval: neglected algebraic corrections to the <outer expansion> and <inner expansion> remain.
\Image[/past-exam-of-the-differential-equations-2-course-of-the-university-of-oxford/2021-boundary-layer.png]
{title=Leading composite approximation with a boundary layer at the right endpoint}
{description=Original sketches for $\epsilon=0.04$ and $0.08$. The dashed curve is the <outer solution> $e^{x-1}$; the inset plots the leading <inner expansion> $e(1-e^{-2X})$ against $X=(2-x)/\epsilon$.}
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