Consider the differential equation
$$
x^{3} y^{\prime \prime}+\alpha x y^{\prime}+\beta y=0
$$
where \(\alpha\) and \(\beta\) are real constants and \(\alpha \neq 0\).
(a) Show that \(x=0\) is an irregular singular point.
(b) By attempting to determine a solution of the form \(\sum_{n=0}^{\infty}
a_{n} x^{r+n},\) show that the indicial equation for \(r\) is linear, and
consequently there is only one formal solution of the assumed form.
(c) Show that if \(\beta / \alpha=-1,0,1,2, \ldots,\) then the formal series
solution terminates and therefore is an actual solution. For other values of
\(\beta / \alpha\) show that the formal series solution has a zero radius of
convergence, and so does not represent an actual solution in any interval.