1. (a) Let S(x), P(x), Q(x) be polynomials with no common factors. Give the condition on S(x), which distinguishes between x being an ordinary or a singular point for the equation S(x)y" + P(x)y' + Q(x)y = 0. If x is a singular point, give the conditions for it to be regular. (1) Locate the singular points of the equation (1 - x²)2 d²y dy - dx2 1) +(x+1)y = 0, dx (2) and determine whether they are regular or irregular. (b) Show that x = O is a regular singular point for the equation (Bessel's equation of zero order) xy" + y + xy = 0 and find the corresponding indicial equation. Find a series solution of the form y=1+ Σana", n=1 deriving the general recurrence relation and calculating the explicit form of an. Show that a2k-1 = 0, k = 1, 2,.... Determine first four nonzero terms of the expansion. (c) Justify the statement: If equation (1) has a solution 1 = xn, n2, then the point x = 0 is singular. In other words, x = 0 cannot be an ordinary point of the equation.