= Solution
The negative drift moves the <outflow boundary layer> to $x=1$. The first-order outer equation is $-(1+x)y_0'+y_0=0$. It must satisfy the left <boundary condition>, giving $y_0=1+x$. This tends to two at the right endpoint.
Use $X=(1-x)/\epsilon$. The leading inner equation is $Y_0''+2Y_0'=0$, and impose $Y_0(0)=1$ and $Y_0\to2$ on matching. Thus $Y_0=2-e^{-2X}$, and a leading <uniform asymptotic approximation> is
$$
\boxed{y\sim1+x-e^{-2(1-x)/\epsilon}}.
$$
For higher accuracy, expand the drift as $2-\epsilon X$ in the inner equation, solve the successive inhomogeneous equations, and match to outer corrections satisfying zero left endpoint data. There is no need for a left endpoint layer. Choosing the growing exponential at the wrong endpoint would prevent matching and is the reason the layer location matters.
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