Consider the hypergeometric equation where a, b, c are constants. x(1-x)y" (c-(a+b+1)x)y' - aby = 0 E (a) Show x = 0 is a regular singular point of (1. Show that the indicial equation has roots r₁ = 0 and r21 C. (b) Assume that r2 is not a positive integer. Then we are guaranteed to have a Frobenius series solution for r₁ = 0. Find the recursion relation for this solution and find the first 3 three non-trivial terms (i.e. a1, a2, a3). These are the first three terms of the hypergeometric series F(a, b, c, x). By using the ratio test and the recursion relation deduce that the solution will converge for a <1 - hence F(a, b, c, x) defines an analytic function in this interval. (c) Show that x = 1 is also a regular singular point of the hypergeometric equation (1). In fact there is a third regular singular point ‘at infinity'. To show this, let w = 1/x and rewrite the hypergeometric differential equation in terms of the new independent variable w. Show that w = 0 is a regular singular point. It can be proved that any 2nd order ODE with three regular singular points can be written in the form (1.