Question

question 2 part b please with explanations. thank you!

2. Find a particular solution to each nonhomogeneous recurrence (a) an - 5an-1 – 6an-2 = 2" (for n > 2) (b) an – an-1 – 6an-2 = 5.3" (for n > 2) (C) an - An-1 – 6an-2 = n (for n > 2)

Answer 1

. Given non homogeneous recurrence relation is, Given non 9n -an-, -6.4n-2 = 5.9 Associated homogeneous recurrence relation is in quanny - 696-2 =0 @ Suppose ona to o ancia en Por2 = 2n-2 * Bano become : 2n_ anni 621-2 = 0 = 24-2722-2-6)=0 = 22-2-6=0 - This is characteristic equation corresponding to ean @ We find its root, 22-d26= (2-3) (2+2) 20 -3+2 => 2=3,-2 2 - 3, Le 5-2 - Solution of eq" ☺ is, an= A x? + Aa La Tan = A, 31 + A2 (-2;9 ) --- - 6 :: Eq" is solution of homegeneous equation ®

We find particular solution by two ways o using formula in Here fins a 5x on We know if f(n)= koeltĄ Kian & 2²7 para be Characteristic equation of associated homogeneous equation Crecurrence relation) & Q,, da be roots of auxillary equation W IF 2=2, or L=La then particular solution is op te Boan a Bn.3" si anat = B(n-1).<n's BCN-1) 39-! peta z = BCN-2) 34-2 . Eano become Br BP - B(n-1) 3"1_ 6B(0-2) 80-2 = 5x8n => 30-27 BnXge – B(N-1)3 - 6B(A-2)) = 5x3+) o [ 9 Bn - BBO + 3B -- 6Bn+126) - 5x32 g. 15B=45 B=3) 1. Opt a 30.3" 0.3n+! ร . is particular solution

or © To find particular solution leta + Bxgn be particular solution o contine a Byzny, a 2 a Bx31-2 .: Eano become, (Bx3")_(B x31-1)=6 B230-2) = 5x35 31-2[(8x32) (1343) - 6B] = 5*39 3 932 - 38 – 6B = 5x32 0 = 5x32 we cann't find value B. Here - our guess fails Next Suppose ao Bn3 be particular solution as andia B(n-1) 30-1, an a BCN-2)..gnr2 Ean o become - Bron - BCR-1) 81_ 6B (0-2) gn=2 5600 30-2 [ Bn 32 - BPO-1)3 - 6B * (0-2)) = 5x31 a) IBD - 3BC1-12 - 6 BCN-2) – 54932 ng 9Bn - 3B0 +386 Br +128 - 5x9 153 = 5x9 1B=3 lant 30.30] is particular solution