= Solution
{c}
In a layer of thickness $\ell\ll1$, the third-<derivative> and first-<derivative> terms have sizes $\epsilon\ell^{-3}y$ and $\ell^{-1}y$. Their <dominant balance> requires $\ell^2\sim\epsilon$, so
$$
\boxed{\ell=O(\sqrt\epsilon).}
$$
The local fast modes are exponentials with rates $\pm1/\sqrt\epsilon$. A mode decaying into the interval can occur at either endpoint: $e^{-x/\sqrt\epsilon}$ at zero and $e^{-(1-x)/\sqrt\epsilon}$ 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.
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