The function $y(x)$ satisfies

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

What does it mean to say that the point $x=0$ is (i) an ordinary point and (ii) a regular singular point of this differential equation? Explain what is meant by the indicial equation at a regular singular point. What can be said about the nature of the solutions in the neighbourhood of a regular singular point in the different cases that arise according to the values of the roots of the indicial equation?

State the nature of the point $x=0$ of the equation

$$x y^{\prime \prime}+(x-m+1) y^{\prime}-(m-1) y=0$$

Set $y(x)=x^{\sigma} \sum_{n=0}^{\infty} a_{n} x^{n}$, where $a_{0} \neq 0$, and find the roots of the indicial equation.

(a) Show that one solution of $(*)$ with $m \neq 0,-1,-2, \cdots$ is

$$y(x)=x^{m}\left(1+\sum_{n=1}^{\infty} \frac{(-1)^{n} x^{n}}{(m+n)(m+n-1) \cdots(m+1)}\right)$$

and find a linearly independent solution in the case when $m$ is not an integer.

(b) If $m$ is a positive integer, show that $(*)$ has a polynomial solution.

(c) What is the form of the general solution of $(*)$ in the case $m=0$ ? [You do not need to find the general solution explicitly.]