First-order composite expansion for a positive drift ($y_{\mathrm{comp}}=y_{\mathrm{out}}+y_{\mathrm{in}}-y_{\mathrm{overlap}}$)

For $ϵy^{\prime \prime }+a(x)y^{\prime }+b(x)y=0$ with smooth $a$ bounded positively away from zero, the first-order equation for the outer expansion is selected by the right boundary condition. The left mismatch is repaired on the scale $\xi =x/ϵ$. If $a(0)=a_0>0$, the leading inner transient is $B{e}^{-a_0\xi }$. Its next equation is $Y_1^{\prime \prime }+a_0{Y}_1^{\prime }=-a^{\prime }(0)\xi Y_0^{\prime }-b(0)Y_0$. Matching its constant and linear large-$\xi$ terms to the outer expansion permits an order-$ϵ$ composite approximation. A sign reversal of the drift places the transient at the opposite endpoint.