 Gaussian interior layer of a conservative drift equation

The equation $ϵy^{\prime \prime }+x{y}^{\prime }+y=0$ integrates to $ϵy^{\prime }+xy=C$. Equal endpoint data $y(-1)=y(1)=A$ force $C=0$ and give the unique Gaussian solution above. Its central width is $\sqrt{ϵ}$, with amplitude $A{e}^{1/(2ϵ)}$. The exponentially large central mode is missed by an assumed bounded algebraic outer expansion $C/x$. Changes of order one relative to either endpoint value occur over distance $O(ϵ)$ from that endpoint.