Consider the following differential equation. 6y" + (x + 1)y + 3y = 0, x0 = 2 (a) Seek a power series solution for the given differential equation about the given point xo. O y = do + al (x - 2) - (- (3do + 301) (n+1)an-2 + 3(n -1.)an-1>(x - 2) + ... 12 -) (x - 2)+ ... - ( - 6(n)(n - 1) O y = an + al (x - 2) - - _ Sao (x - 2) + ... - (n + 3)an (x - 2) + ... 12 6(n)(n - 1) O y = do + al (x - 2) - (- 3do + 301 ) (x - 2) + ... - ( n + 3 ) an + 3 (n + 1 )an+1 ) (x - 2) + ... 2 (n)(n - 1) O y = do + aix -12 3do + 301 2 4 - {n + 3)an + 3(n + 1)anth ) x" +.. 6(n)(n - 1) O 12 y = do + al ( x - 2) - ( 50 0 + 21 ) ( x - 2)2 + ... _ ((n + 3)an + 2(n + 1)an+1) (x - 2)" + ... 6(n)(n - 1) Hint (b) Find the recurrence relation. antz = _ (n - 3)an - 4(n + 1)an+1 12 (n + 2) (n + 6)a, + 5(n + 1)a,+1 3 (n + 2) (n + 1) (n + 3) a, + 12(n + 1 )an+1 (n + 2)(n + 1) (n - 3)an + (n + 1)an+1 6(n + 2)(n + 1) O ant2 = (n + 3)an + 3(n+ 1)a,+1 6(n + 2)(1 + 1)