The Bessel equation of order one is x2 y__ + xy_ + ( x2 1) y = 0. a. Show that x = 0 is

QUESTION:

The Bessel equation of order one is x2 y__ + xy_ + ( x2 1) y = 0. a. Show that x = 0 is a regular singular point. b. Show that the roots of the indicial equation are r1 = 1 and r2 = 1. c. Show that one solution for x > 0 is J1( x) = x 2 _ n=0 (1)n x2n (n + 1)! n! 22n . The function J1 is known as the Bessel function of the first kind of order one. d. Show that the series for J1( x) converges for all x. e. Show that it is impossible to determine a second solution of the form