Exponential-reaction terminal boundary layer ($y_c=-\log (e^{-A}-x)+\log (e^{-A}-1)e^{(x-1)/ϵ}$)

For $ϵy^{\prime \prime }-y^{\prime }+e^y=0$ with $y(0)=A<0$ and $y(1)=0$, the outer solution is $-\log (e^{-A}-x)$. The right-end boundary layer has width $ϵ$ and leading correction $\log (e^{-A}-1)e^{(x-1)/ϵ}$. This follows by setting $X=(x-1)/ϵ$, solving $Y^{\prime \prime }-Y^{\prime }=0$, and matching its constant part. The leading correction changes sign at $A=-\log 2$.