From Griffiths, if we have some potential $V(x)$ and energy $E$ such that $E=V(0)$ where $V(x)<E$ for all $x<0$ and $V(x)>E$ for all $x>0$. In the patching region, Griffiths uses only one of the asymptotic Airy functions: $$\psi_{patch} = a\mathrm{Ai}(\alpha x)$$ instead of the general linear combination of both Airy functions. What is the justification behind this? I tried deriving the connection formulas without ignoring the second airy function, and I end up with two independent equations with 3 unknowns so I'm not sure why Griffiths is allowed to simply 'choose' his patching function.
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