Consider the laplace equation for a ball of radius r described in spherical coordinates (r, 0) 2 1 cot 72 0= n where is the zenith angle and assume u is independent on the azirnuth angle d. a) by separation of variables, derive two ordinary differential equations of r and w: = cos e given by 2 f" (r) +2rf (r) - n(n + 1) f, (r) = 0, (1 w2)g (w) - 2wg", (w) +n(n +1)g, (w) 0. (n 0,1,2,. .) b) find fn (r) and gn (w) satisfying g(1) = 1 for n = 1,3. c) show that for n m, gn (w)gm (w)dw = 0. d) assume the boundary condition is given by u(r, o) = cos 0. then determine u(r, ) inside the ball.