Show that the equation

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

has an irregular singular point at infinity. Using the Liouville-Green method, show that one solution has the asymptotic expansion

$$y(x) \sim \frac{1}{x} e^{x}\left(1+\frac{1}{2 x}+\ldots\right)$$

as $x \rightarrow \infty$