(a) Classify the nature of the point at $\infty$ for the ordinary differential equation

$$y^{\prime \prime}+\frac{2}{x} y^{\prime}+\left(\frac{1}{x}-\frac{1}{x^{2}}\right) y=0 .$$

(b) Find a transformation from $(*)$ to an equation of the form

$$u^{\prime \prime}+q(x) u=0$$

and determine $q(x)$.

(c) Given $u(x)$ satisfies ( $\dagger$, use the Liouville-Green method to find the first three terms in an asymptotic approximation as $x \rightarrow \infty$ for $u(x)$, verifying the consistency of any approximations made.

(d) Hence obtain corresponding asymptotic approximations as $x \rightarrow \infty$ of two linearly independent solutions $y(x)$ of $(*)$.