 Solution

In a layer of thickness $\ell \ll 1$, the third-derivative and first-derivative terms have sizes $ϵ{\ell }^{-3}y$ and ${\ell }^{-1}y$. Their dominant balance requires ${\ell }^2\sim ϵ$, soThe local fast modes are exponentials with rates $\pm 1/\sqrt{ϵ}$. A mode decaying into the interval can occur at either endpoint: $e^{-x/\sqrt{ϵ}}$ at zero and $e^{-(1-x)/\sqrt{ϵ}}$ at one. The zero fast exponent represents the matching outer component. The term $xy$ is lower order on either fast scale, and even smaller near zero; it creates no further independent layer thickness.