Boundary layer around a moving endpoint ($z=(x-\epsilon )/{\epsilon }^2$)

For $\epsilon x{w}^{\prime \prime }+w^{\prime }+2xw=0$ with left endpoint $x=\epsilon$, diffusion there is of size ${\epsilon }^2$. The distinguished coordinate is $z=(x-\epsilon )/{\epsilon }^2$, giving $(1+\epsilon z)W^{\prime \prime }+W^{\prime }+2{\epsilon }^3(1+\epsilon z)W=0$. Matching to one with zero left value gives $W=1-e^{-z}-\epsilon z(z+2)e^{-z}/2+O({\epsilon }^2)$. The layer width is different from the endpoint's distance from zero.