(a) Outline the Liouville-Green approximation to solutions $w(z)$ of the ordinary differential equation

$$\frac{d^{2} w}{d z^{2}}=f(z) w$$

in a neighbourhood of infinity, in the case that, near infinity, $f(z)$ has the convergent series expansion

$$f(z)=\sum_{s=0}^{\infty} \frac{f_{s}}{z^{s}}$$

with $f_{0} \neq 0$.

In the case

$$f(z)=1+\frac{1}{z}+\frac{2}{z^{2}},$$

explain why you expect a basis of two asymptotic solutions $w_{1}(z), w_{2}(z)$, with

$$\begin{aligned} &w_{1}(z) \sim z^{\frac{1}{2}} e^{z}\left(1+\frac{a_{1}}{z}+\frac{a_{2}}{z^{2}}+\cdots\right), \\ &w_{2}(z) \sim z^{-\frac{1}{2}} e^{-z}\left(1+\frac{b_{1}}{z}+\frac{b_{2}}{z^{2}}+\cdots\right), \end{aligned}$$

as $z \rightarrow+\infty$, and show that $a_{1}=-\frac{9}{8}$.

(b) Determine, at leading order in the large positive real parameter $\lambda$, an approximation to the solution $u(x)$ of the eigenvalue problem:

$$u^{\prime \prime}(x)+\lambda^{2} g(x) u(x)=0 ; \quad u(0)=u(1)=0$$

where $g(x)$ is greater than a positive constant for $x \in[0,1]$.