The function satisfies
What does it mean to say that the point 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 of the equation
Set , where , and find the roots of the indicial equation.
(a) Show that one solution of with is
and find a linearly independent solution in the case when is not an integer.
(b) If is a positive integer, show that has a polynomial solution.
(c) What is the form of the general solution of in the case ? [You do not need to find the general solution explicitly.]