 Solution

Set $s=x+1\in [0,2]$. The drift now vanishes quadratically at the left endpoint, and the outer expansion selected from $s=2$ is proportional to $e^{1/s}$, which becomes singular as $s\downarrow 0$. The oscillatory endpoint with a quadratically vanishing drift is not an ordinary monotone layer of width ${ϵ}^{1/3}$: balancing just diffusion and drift on that scale neglects the zeroth-order $y$ term.

The formal local exponential rates obey $ϵ{\lambda }^2+s^2\lambda +1=0$. For $s\gg {ϵ}^{1/4}$ their slow and fast branches are approximately $-s^{-2}$ and $-s^2/ϵ$. They coalesce when $s^4\sim 4ϵ$. This locates the main transition a distance $O({ϵ}^{1/4})$ from the left endpoint.

A reliable procedure removes the first derivative exactly:Use WKB approximation for this transformed equation on each side of its unique positive zero $s_t=\sqrt{2}{ϵ}^{1/4}+O({ϵ}^{1/2})$. For $s>s_t$ the transformed modes are exponential; for $0<s<s_t$ they are oscillatory. Their physical amplitudes include the displayed exponential prefactor.

At $s_t$, $P^{\prime }(s_t)=O({ϵ}^{3/4})$, so the Airy turning-point connection formula usesThis layer connects the oscillatory and exponential WKB approximation branches. On the transition scale $s={ϵ}^{1/4}S$, the exact equation is ${ϵ}^{1/2}{v}_{SS}=(S^4/4-1+{ϵ}^{1/4}S)v$, displaying a small effective WKB approximation parameter ${ϵ}^{1/4}$.

Close to $s=0$, the local wavelength is $O(\sqrt{ϵ})$: setting $s=\sqrt{ϵ}\xi$ gives $Y_{\xi \xi }+Y+\sqrt{ϵ}{\xi }^2{Y}_{\xi }=0$, whose leading solutions are sine and cosine. This endpoint form imposes $y(-1)=1$. Within the oscillatory region, the prefactor changes by order one over the additional envelope scale $s=O({ϵ}^{1/3})$. The $\sqrt{ϵ}$ wavelength and ${ϵ}^{1/3}$ envelope are subscales of the same oscillatory region, not additional omitted ordinary layers.

Use the outer exponential/regular branches for $s\gg {ϵ}^{1/4}$, oscillatory WKB approximation inside the $O({ϵ}^{1/4})$ transition region, an $O({ϵ}^{5/12})$ Airy neighborhood of $s_t$, and the $O(\sqrt{ϵ})$ local endpoint form to apply the left boundary data. Match their constants and enforce the right boundary data at $s=2$; there is no separate right endpoint layer required. Matching amplitudes need not remain $O(1)$, and parameter values admitting a homogeneous Dirichlet solution require a separate solvability check rather than an assumed uniform algebraic expansion.