 Solution

The negative drift moves the outflow boundary layer to $x=1$. The first-order outer equation is $-(1+x)y_0^{\prime }+y_0=0$. It must satisfy the left boundary condition, giving $y_0=1+x$. This tends to two at the right endpoint.

Use $X=(1-x)/ϵ$. The leading inner equation is $Y_0^{\prime \prime }+2Y_0^{\prime }=0$, and impose $Y_0(0)=1$ and $Y_0\to 2$ on matching. Thus $Y_0=2-e^{-2X}$, and a leading uniform asymptotic approximation isFor higher accuracy, expand the drift as $2-ϵX$ 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.