Derive the leading-order Liouville Green (or WKBJ) solution for $\epsilon \ll 1$ to the ordinary differential equation

$$\epsilon^{2} \frac{d^{2} f}{d y^{2}}+\Phi(y) f=0$$

where $\Phi(y)>0$.

The function $f(y ; \epsilon)$ satisfies the ordinary differential equation

$$\epsilon^{2} \frac{d^{2} f}{d y^{2}}+\left(1+\frac{1}{y}-\frac{2 \epsilon^{2}}{y^{2}}\right) f=0$$

subject to the boundary condition $f^{\prime \prime}(0)=2$. Show that the Liouville-Green solution of (1) for $\epsilon \ll 1$ takes the asymptotic forms

where $\alpha_{1}, \alpha_{2}, B$ and $\theta_{2}$ are constants.

$\left[\right.$ Hint: You may assume that $\left.\int_{0}^{y} \sqrt{1+u^{-1}} d u=\sqrt{y(1+y)}+\sinh ^{-1} \sqrt{y} \cdot\right]$

Explain, showing the relevant change of variables, why the leading-order asymptotic behaviour for $0 \leqslant y \ll 1$ can be obtained from the reduced equation

$$\frac{d^{2} f}{d x^{2}}+\left(\frac{1}{x}-\frac{2}{x^{2}}\right) f=0$$

The unique solution to $(2)$ with $f^{\prime \prime}(0)=2$ is $f=x^{1 / 2} J_{3}\left(2 x^{1 / 2}\right)$, where the Bessel function $J_{3}(z)$ is known to have the asymptotic form

$$J_{3}(z) \sim\left(\frac{2}{\pi z}\right)^{1 / 2} \cos \left(z-\frac{7 \pi}{4}\right) \text { as } z \rightarrow \infty .$$

Hence find the values of $\alpha_{1}$ and $\alpha_{2}$.

$$\begin{aligned} & f \sim \alpha_{1} y^{\frac{1}{4}} \exp (2 i \sqrt{y} / \epsilon)+\alpha_{2} y^{\frac{1}{4}} \exp (-2 i \sqrt{y} / \epsilon) \quad \text { for } \quad \epsilon^{2} \ll y \ll 1 \\ & \text { and } \quad f \sim B \cos \left[\theta_{2}+(y+\log \sqrt{y}) / \epsilon\right] \quad \text { for } \quad y \gg 1, \end{aligned}$$