= Solution
{c}
In the <overlap region> $\epsilon^{1/3}\ll x\ll1$, write $X=x/\epsilon^{1/3}$. The supplied <Airy function> asymptotics give
$$
Y\sim\frac{\alpha\epsilon^{1/12}}{2\sqrt\pi}x^{-1/4}
e^{-2x^{3/2}/(3\sqrt\epsilon)}
+\frac{\beta\epsilon^{1/12}}{\sqrt\pi}x^{-1/4}
e^{2x^{3/2}/(3\sqrt\epsilon)}.
$$
The outer prefactor $e^{-x^2/4+x/2}$ tends to one in this overlap. Therefore
$$
c_1=\frac{\alpha\epsilon^{1/12}}{2\sqrt\pi},\qquad
c_2=\frac{\beta\epsilon^{1/12}}{\sqrt\pi}.
$$
The recessive condition $c_2=0$ gives $\beta=0$ and $\alpha=1/a_0=3^{2/3}\Gamma(2/3)$. Hence the leading inner answer is
$$
\boxed{y_{\rm in}(x)=\frac{\operatorname{Ai}(x/\epsilon^{1/3})}{\operatorname{Ai}(0)}.}
$$
Away from the turning point, the matched <WKB approximation> is
$$
\boxed{y(x)\sim
\frac{\epsilon^{1/12}}{2\sqrt\pi\,\operatorname{Ai}(0)}
x^{-1/4}\exp\left[-\frac{2x^{3/2}}{3\sqrt\epsilon}-\frac{x^2}4+\frac x2\right].}
$$
A useful <multiplicative composite expansion> through the turning region is $e^{-x^2/4+x/2}\operatorname{Ai}(x/\epsilon^{1/3})/\operatorname{Ai}(0)$: it is finite and normalized at zero and matches the displayed outer expression for fixed positive $x$. It is a small-$\epsilon$ approximation across the turning point, not a claim of relative uniform accuracy out to arbitrarily large $x$.
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