What is the general form of the particular solution of the linear nonhomogeneous recurrencerelation an = -6an-1-−6ªn-1 — 8ªn-2 + F(n) if(a) F(n) = n²(b) F(n) = (nª + 1)4″

Given nonhomogeneous equation isan=−6an−1−8an−2+F(n)Then the corresponding homogeneous equation…

Answered: What is the general form of the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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Consider the following linear non-homogeneous recurrence relation where go = 1, g1 = 3 and gn†1=-9n + 6gn-1 - 3 for n > 1. The reduced closed form of gn can be represented as gn = (Px p" + Q x q" + C') where p and q are the solutions of the characteristic equation of the recurrence. Given that, p is smaller than q. Here, P,Q,C, A are all integers. What is the value of p? What is the value of P?? What is the value of C? What is the value of A?
nan (a – 3)n-1 + x Σ an (x - 3)" = 0, - n=0 then the recurrence relation can be written as (n + 1)an+1 – an−1 + 3an = 0, (n + 1)an+1 + an−1 + an=0, – (n+1)an+1 + an−1 + 3an = 0, η>1 (n + 2)an+2 + an−1 + an = 0, (n + 2)an+2 + an−1 + an = 0, - Q6. 11 Σ n=1 = (a) αι = =do, (b) αι = 3a0 , (c) a1 = - –3ao, (d) a1 = do , (e) a1 = =do , n21 n>1 η21 η>1
What is the general form of the particular solution guaranteed to exist by Theorem 6 of the linear nonhomogeneous recurrence relation an= 6an-1-12an -2 +8an – 3 + F(n) if F(n) = n²? Multiple Choice O O O P2n³+ p₁n² + pon (p2n³ + p₁n² + Pon)2n (p2n² + p₁n+po)2n p2n² + pin + po
3. Find the recurrence relation for y" + 3ry – y = 0. Use the initial conditions y(0) = 2 and y (0) = 0 to find the values the first four non-zero coefficients in y = n=0
7. Consider the second-order homogeneous recurrence relation an 5an-1 6an-2 with initial conditions ao : 0, a1 = 5. (a) Find the next three terms of the sequence. (b) Find the general solution. (c) Find the unique solution with the given initial conditions.
What is the form of the particular solution of the linear non-homogeneous recurrence relation. [hint: the characteristic equation factorised is (x+5)²(x+1)²] an = -12an-1-36an-2 - 60an-3-25an-4+ (3n³ + 1) (-5)" - 1 −
Starting with P0(x) = 1 and P1(x) = x, use the recurrence relation (n+1)Pn+1+nPn−1 =(2n+1)xPn, n =1,2,3,... to determine P2, P3, and P4.
2. (a) Let (F(n)n>o be a sequence of real numbers and let k E N. Explain what it means for (F(n))n>o to satisfy a (k + 1)-term recurrence relation with constant coefficients. (b) Suppose (F(n))n>0 satisfies the following recurrence relation: F(n) = aF(n – 1) for n>1 and F(0) = b, where a, b e N. Compute the values of F(1), F(2), and then find a general formula for F(n) for arbitrary n > 0. (c) If (F(n))n>0 is a sequence, write down the generating function for (F(n))n>0- (d) A sequence (F(n))n>0 has the following initial terms F(0) = 2, F(1) = 4, F(2) = 10 and satisfies the recurrence relation F(n) = 3F(n – 1) – 2 for n > 1. 1 (i) Show that this sequence also satisfies the relation F(n) = 4F(n – 1) – 3F(n – 2) for n> 2. (ii) Using the generating functions approach, find a closed form for F(n) for all n> 0.
Consider the nonhomogeneous linear recurrence relation a, = 3a,-1 + 2" %3D (a) Show that an -2"+1 is a solution of this recurrence relation. (b) Find all the solutions of this recurrence relation. (c) Find the solution with ao = 1.
Solve the following second-order linear recurrences: a) T(n) = T(n - 2) for n > 2, T(0) = 5 and T(1) = -1. b) T(n) = -4T(n – 1) + 5T(n – 2) for n 2 2, T(0) = 2 and T(1) = 8. %3D
Solve the recurrence relation (find the closed form) defined by T(1) = 1 T(2) = 14 T(n) = 3T(n-1) + 4T(n-2), n ≥ 3
Use the Laplace transform to solve the given initial-value problem. y" - 3y' = 8e²t - 2e-t, y(0) = 1, y'(0) = -1 y(t) 11

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What is the general form of the particular solution of the linear nonhomogeneous recurrence relation an -6an-1-8an-2 + F(n) if (a) F (n) = n² (b) F(n) = (nª + 1)4″