= Endpoint derivative in a negative-drift boundary layer
{title2=$y'(1)=-2/\epsilon+4+O(\epsilon)$}
For $\epsilon y''-(1+x)y'+y=0$, $y(0)=1$, $y(1)=1+\epsilon$, the exact outer solution is $1+x$ and the mismatch is at the right endpoint. In $X=(1-x)/\epsilon$, matching gives $Y_0=2-e^{-2X}$ and $Y_1=-X+(1-X-X^2/2)e^{-2X}$. Differentiation at $X=0$ gives the displayed endpoint slope. The order-one term requires the first inner correction; differentiating only the leading boundary layer loses it.
Back to article page
