 Square-root boundary layer at a vanishing drift

In $\epsilon y^{\prime \prime }+2x{y}^{\prime }-2xy=0$, the drift vanishes at the endpoint $x=0$. Balancing diffusion against drift gives the inner expansion coordinate $z=x/\sqrt{\epsilon }$. The leading layer equation is $Y^{\prime \prime }+2z{Y}^{\prime }=0$, with a bounded error function profile. A regular outer first-order calculation alone cannot impose both endpoint values.