Consider the differential equation

$$\frac{d^{2} w}{d x^{2}}=q(x) w$$

where $q(x) \geqslant 0$ in an interval $(a, \infty)$. Given a solution $w(x)$ and a further smooth function $\xi(x)$, define

$$W(x)=\left[\xi^{\prime}(x)\right]^{1 / 2} w(x) .$$

Show that, when $\xi$ is regarded as the independent variable, the function $W(\xi)$ obeys the differential equation

$$\frac{d^{2} W}{d \xi^{2}}=\left\{\dot{x}^{2} q(x)+\dot{x}^{1 / 2} \frac{d^{2}}{d \xi^{2}}\left[\dot{x}^{-1 / 2}\right]\right\} W$$

where $\dot{x}$ denotes $d x / d \xi$.

Taking the choice

$$\xi(x)=\int q^{1 / 2}(x) d x$$

show that equation $(*)$ becomes

$$\frac{d^{2} W}{d \xi^{2}}=(1+\phi) W$$

where

$$\phi=-\frac{1}{q^{3 / 4}} \frac{d^{2}}{d x^{2}}\left(\frac{1}{q^{1 / 4}}\right)$$

In the case that $\phi$ is negligible, deduce the Liouville-Green approximate solutions

$$w_{\pm}=q^{-1 / 4} \exp \left(\pm \int q^{1 / 2} d x\right)$$

Consider the Whittaker equation

$$\frac{d^{2} w}{d x^{2}}=\left[\frac{1}{4}+\frac{s(s-1)}{x^{2}}\right] w$$

where $s$ is a real constant. Show that the Liouville-Green approximation suggests the existence of solutions $w_{A, B}(x)$ with asymptotic behaviour of the form

$$w_{A} \sim \exp (x / 2)\left(1+\sum_{n=1}^{\infty} a_{n} x^{-n}\right), \quad w_{B} \sim \exp (-x / 2)\left(1+\sum_{n=1}^{\infty} b_{n} x^{-n}\right)$$

as $x \rightarrow \infty$.

Given that these asymptotic series may be differentiated term-by-term, show that

$$a_{n}=\frac{(-1)^{n}}{n !}(s-n)(s-n+1) \ldots(s+n-1) \text {. }$$