Consider the differential equation

$$\tag{†} y^{\prime \prime}-y^{\prime}-\frac{2(x+1)}{x^{2}} y=0$$

(i) Classify what type of regularity/singularity equation $(†)$ has at $x=\infty$.

(ii) Find a transformation that maps equation ($†$) to an equation of the form

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

(iii) Find the leading-order term of the asymptotic expansions of the solutions of equation $(*)$, as $x \rightarrow \infty$, using the Liouville-Green method.

(iv) Derive the leading-order term of the asymptotic expansion of the solutions $y$ of ($†$). Check that one of them is an exact solution for $(†)$.