The differential equation

$$f^{\prime \prime}=Q(x) f$$

has a singular point at $x=\infty$. Assuming that $Q(x)>0$, write down the Liouville Green lowest approximations $f_{\pm}(x)$ for $x \rightarrow \infty$, with $f_{-}(x) \rightarrow 0$.

The Airy function $\operatorname{Ai}(x)$ satisfies $(*)$ with

$$Q(x)=x$$

and $\operatorname{Ai}(x) \rightarrow 0$ as $x \rightarrow \infty$. Writing

$$\operatorname{Ai}(x)=w(x) f_{-}(x)$$

show that $w(x)$ obeys

$$x^{2} w^{\prime \prime}-\left(2 x^{5 / 2}+\frac{1}{2} x\right) w^{\prime}+\frac{5}{16} w=0$$

Derive the expansion

$$w \sim c\left(1-\frac{5}{48} x^{-3 / 2}\right) \quad \text { as } \quad x \rightarrow \infty$$

where $c$ is a constant.