= Solution
The reduced outer equation is $(1+x)y_0'+y_0=0$. Imposing the right boundary condition gives
$$
\boxed{y_{\rm out}=\frac2{1+x}
+\epsilon\left[\frac2{(1+x)^3}-\frac1{2(1+x)}\right]+O(\epsilon^2).}
$$
The condition at $x=0$ requires $X=x/\epsilon$. The first two inner terms are
$$
\boxed{Y_0=2-e^{-X},
\qquad
Y_1=-2X+\frac32+\left(\frac{X^2}{2}-\frac32\right)e^{-X}.}
$$
They match $2+\epsilon(-2X+3/2)$. The <additive composite expansion> is
$$
\boxed{
y_{\rm comp}=\frac2{1+x}
+\epsilon\left[\frac2{(1+x)^3}-\frac1{2(1+x)}\right]
-e^{-x/\epsilon}
+\epsilon\left[\frac12\left(\frac x\epsilon\right)^2-\frac32\right]e^{-x/\epsilon}.}
$$
On $[-1,1]$, the coefficient $1+x$ vanishes at the left endpoint. The ordinary $O(\epsilon)$ exponential layer is replaced by a turning-point endpoint region of width $O(\sqrt\epsilon)$, where all three terms in the equation enter the leading balance.
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