Obtain an asymptotic representation of the solution of Airy's equation $$\frac{d^2y}{dz^2}-zy=0$$ for $|z|$ large using the change of variable $z=\epsilon^{-2/3}\xi$.
The change of variable leads to an equation $$\frac{d^2y}{d\xi^2}-\epsilon^{-2}\xi y=0.$$ Setting $K=\epsilon^{-2}$ and $q(\xi)=-\xi$ the equation is in WKB form with solution $$y=A_+q(\xi)^{-1/4} e^{iK\int \sqrt{q(\xi)}d\xi}+A_-q(\xi)^{-1/4} e^{-iK\int \sqrt{q(\xi)}d\xi} \\ =A_+(-\xi)^{-1/4} e^{i\epsilon^{-2}\frac23(-\xi)^{3/2}}+A_-(-\xi)^{-1/4} e^{-i\epsilon^{-2}\frac23(-\xi)^{3/2}}$$ My problem is how to get the solution into a form which involve only $z$ and $y$ as I don't to see how to get rid of $\epsilon$ . Do I need to use a different $K$ and $q$? Setting $K=\epsilon^{-4/3}$ and $q(\xi)=\epsilon^{-2/3}\xi$ would give a term $(-z)^{-1/4}$ for $q(\xi)^{-1/4}$ for example.