Solve the following sets of recurrence relations and initial conditions: **Problem 6:** Solve the recurrence relation given by:\[ S(k) - S(k-1) - 6S(k-2) = -30 \]with the initial conditions:\[ S(0) = 7, \quad S(1) = 10 \]In this problem, we are tasked with finding a sequence $ S(k) $ that satisfies the linear recurrence relation. Given initial values provide a starting point to determine subsequent terms in the sequence. The relation shows how each term in the sequence depends linearly on previous terms, specifically the two preceding ones, with a constant impact subtracted from each computed term. The task is to find a general expression for $ S(k) $ or to compute specific terms using these conditions.

b) SK-SK-1-6SK-2=-30 ↪1 S0=7 , S1=10The characteristic equation is r2-r-6=0 ⇒r-3r+2=0 ⇒r=-2,3∵ The homogeneous solution is ShK=A-2K+B3KLet SpK=∝=1 SpK-1=SpK-2=∝∴1 ∝-∝-6∝=-30 ⇒∝=306=5∴ SpK=5Hence SK=A-2K+B3K+5 S0=7⇒7=A+B+5⇒A+B=2 ↪2S1=10⇒10=-2A+3B+5 ⇒-2A+3B=5→32x2 +3 ⇒2A+2B-2A+3B=9+5\ ⇒5B=9⇒B=95∴2 ⇒A=2-95=10-95=15∴ SK=15-2K+953K+5

Answered: 6. S(k) – S(k – 1) – 6S(k – 2) = –30,…

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6. S(k) – S(k – 1) – 6S(k – 2) = –30, S(0) = 7, S(1) = 10

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Solve the following sets of recurrence relations and initial conditions:

**Problem 6:** Solve the recurrence relation given by:

\[ S(k) - S(k-1) - 6S(k-2) = -30 \]

with the initial conditions:

\[ S(0) = 7, \quad S(1) = 10 \]

In this problem, we are tasked with finding a sequence $ S(k) $ that satisfies the linear recurrence relation. Given initial values provide a starting point to determine subsequent terms in the sequence.

The relation shows how each term in the sequence depends linearly on previous terms, specifically the two preceding ones, with a constant impact subtracted from each computed term. The task is to find a general expression for $ S(k) $ or to compute specific terms using these conditions.

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