Question

5y" + (x + 1)x + 6y = 0, X) = 2 (a) Seek a power series solution for the given differential equation about the given point Xo. (n + 4)an-2 + 3(n − 1)2n-1)(x - 2)" +... 600 + 3a1 y = do + 01 (x - 2) - ( -)(x - 2)2 + ... - 10 5(n)(n-1) 600 + 2a1 (n + 6)an + 2(n + 1)an+1 y = ao + a (x - 2)-( -)(x - 2)2 + .. -- -)(x - 2)" + .. 10 5(n)(n-1) бар + Зај (n + 6a, + 3(n + 1)a+1 y = do + a (x - 2)-( -)(x - 2)2 + ... - -)(x - 2)" + ... 2 (n)(n-1) (n+6)an + 3(n + 1)an+1 y = do + ax-( -( -).X" + ... 10 5(n)(n-1) бао (n + 6)an y = do + a (x - 2) - -(x - 2)" +... 10 5(n)(n-1) 600 + 301 )x? + +(x - 2)2 +... (b) Find the recurrence relation. an+2= 2n+2= an+2= (n-6a, + (n + 1)anti 5(n + 2)(n+1) (n + 6)an + 10(n + 1)an+1 (n + 2)(n+1) (n + 5)an +5(n + 1)an+1 6(n + 2)(n + 1) (n+an + 3(n + 1)an+1 5(n + 2)(n+1) (n-6)an - 4(n + 1)an+1 10(n + 2) O an+2 = O an+2

Answer 1

=0 *6=2 is an n=0 The giren differential equation is sy" +(a+) y'+6y= ordinary point of (0) Let the solution of the differential equation (1) be of the form ya Elan (x-2)^. Then dy de la semana de les entre o d²y Therefore the equation (1) transforms to dzy +6y=0 y = an (2-2)" ganun cet x-2=v. dy du 1 an and dre 2 = duz + (v+3) dy duz dv Now MED dy namun int and dy dez {n(n-1) an' yn-2 du no nao d du y and Substituting for day in the equation (2) daze ) neo MED neo 8 we get 5 {n(n.) an una + (0+3) Enamuna +6 Zanun § 5nen-) an bh-2 + nanon + fan anunt ibar neo nel ne2 + 6 anun SO no Ž 5 n(n-1) an un-2 + s(n+6) anavn + Jananus neo n=1 N=2 O

n 5 56n+2)(n+1) antav en=0 +In+b)an va + 3(n+1) amoy ban no no C In the 1st series we replace n by nt2 and in the 3rd series we replace in by nt Ž{scana) chat) 5(n+2)(n+1) an+2 +3(n+) anti + (n+6) ançon neo O This holds only when 5(n+2)(n+1) antz+ 3(n+1) ant+Cnts) an=0 for nyo (n+6) an + 3(n+1) anti Antas 5 cm+iXn+2) → for nyo This is the recurrence relation for the power series solution of the given differential equation, Therefore for the question (6) the 4th option is correct Putting nzo in the recurrence relation, we get saot 3 Az { cate 3 10

n-2 Replacing n by in the recurrence relation, we get s (+4) An-2 + 3(n-1) anay 5 (ni)n an Therefore y an (2-2)" no ao tay (2-2) + a₂(x-2)² + .. Gaothai - dotal + (1-2) (600 (az)? (n+4) anazt3(n-1) 50(my) (122) putting the values of az and an Hence the 1st option is correct (for question