= Solution
Reversing the drift reverses the endpoint carrying the <boundary layer>. The outer equation is $-(1+x)^2y_0'+y_0=0$, and now its condition comes from $x=0$:
$$
y_0(x)=e^{1-1/(1+x)}.
$$
Its value at $x=1$ is $e^{1/2}$, so it cannot by itself satisfy the right boundary value. Put $\xi=(1-x)/\epsilon$ there. The leading inner equation is $Y''+4Y'=0$, with a correction proportional to $e^{-4\xi}$ that decays into the domain. Thus \b[the outer region has size $O(1)$ and the right endpoint layer has width $O(\epsilon/4)=O(\epsilon)$]. One would expand the coefficient near $x=1$, solve the successive equations for the <inner expansion>, match as $\xi\to\infty$, and form a <matched asymptotic expansion> by subtracting the overlap. No turning region occurs because the drift does not vanish on this interval.
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