= First-order composite expansion for a positive drift
{title2=$y_{\rm comp}=y_{\rm out}+y_{\rm in}-y_{\rm overlap}$}
For $\epsilon y''+a(x)y'+b(x)y=0$ 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 $\xi=x/\epsilon$. If $a(0)=a_0>0$, the leading inner transient is $B e^{-a_0\xi}$. Its next equation is $Y_1''+a_0Y_1'=-a'(0)\xi Y_0'-b(0)Y_0$. Matching its constant and linear large-$\xi$ terms to the <outer expansion> permits an order-$\epsilon$ composite approximation. A sign reversal of the drift places the transient at the opposite endpoint.
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