_Ae 4. Use a series solution method to find solutions to thelfoerdrJequation, (r -,’) r” -2*g’ * n(n+ 1);; (a) First verify the indicial equation is k(k -1) : 0 (b) For k :0 (oo 10, 01 : 0), obtain the recursion, aj+2:i( i+1)-n(n+1) u+L)(j+D aj (5) show that the solution diverges for r : +1 if the series continue to infinity, show that an appropriate choice of n truncates the series, and solve the recursion formula to give a compact expression for the Legendre polynomial. (c) For k: | (oo * 0, o1 : 0), obtain the recursion, U + 1111 +2) -n(n + L) uj+2:@aj a. l@):;xvla b. l@): #””p (-ff) ….’.’,.–6. Consider afunction f (r)which vanishes for N values of r for r € (a,b). Denotethe values {*r,*r,…rr}. Show that I,o o* g(r),tr (u)i : * r(“,) [l#1,:, ] -‘ (4) (6) show that the solution diverges for u : t1 if the series continue to infinity, show that an appropriate choice of. n truncates the series, and solve the recursion formula to give a compact expression for the Legendre polynomial. -5-ealc[lateTheToniiei Transfoims on** —q/t As a result, in shorthand it is common to write that d[/(r)] -r d(r -ro)llf,@o)l in the neighborhood of a zero of the function f (*), f (*o) : 0. Use the above result to evaluate the’ integral: (7) to* o* e-t’d(cosr^.,2) (8)