Consider solutions to the equation

$$\frac{d^{2} y}{d x^{2}}=\left(\frac{1}{4}+\frac{\mu^{2}-\frac{1}{4}}{x^{2}}\right) y$$

of the form

$$y(x)=\exp \left[S_{0}(x)+S_{1}(x)+S_{2}(x)+\ldots\right]$$

with the assumption that, for large positive $x$, the function $S_{j}(x)$ is small compared to $S_{j-1}(x)$ for all $j=1,2 \ldots$

Obtain equations for the $S_{j}(x), j=0,1,2 \ldots$, which are formally equivalent to ( $)$. Solve explicitly for $S_{0}$ and $S_{1}$. Show that it is consistent to assume that $S_{j}(x)=c_{j} x^{-(j-1)}$ for some constants $c_{j}$. Give a recursion relation for the $c_{j}$.

Deduce that there exist two linearly independent solutions to $(\star)$ with asymptotic expansions as $x \rightarrow+\infty$ of the form

$$y_{\pm}(x) \sim e^{\pm x / 2}\left(1+\sum_{j=1}^{\infty} A_{j}^{\pm} x^{-j}\right)$$

Determine a recursion relation for the $A_{j}^{\pm}$. Compute $A_{1}^{\pm}$and $A_{2}^{\pm}$.