Wednesday, 10 April 2019

11:13 AM

Determine the particular solution

Given that the general solution of the homogeneous recurrence relation ﷐𝑎﷮𝑛﷯−7﷐𝑎﷮𝑛−1﷯+12﷐𝑎﷮𝑛−2﷯=0 is ﷐ℎ﷮𝑛﷯=𝐴﷐3﷮𝑛﷯+𝐵﷐4﷮𝑛﷯
Guess a formula for a particular solution for the below relations

REMEMBER the guess cannot have common terms with the homogeneous

Recurrence Relation
Guess a Particular Solution
﷐𝑎﷮𝑛﷯−7﷐𝑎﷮𝑛−1﷯+12﷐𝑎﷮𝑛−2﷯=30
constant
﷐𝑝﷮𝑛﷯=𝑐
﷐𝑎﷮𝑛﷯−7﷐𝑎﷮𝑛−1﷯+12﷐𝑎﷮𝑛−2﷯=30𝑛
Polynomial degree one
﷐𝑝﷮𝑛﷯=𝑐𝑛+𝑑 (﷐𝑝﷮𝑛﷯=𝑐𝑛 will create a contradiction later)
﷐𝑎﷮𝑛﷯−7﷐𝑎﷮𝑛−1﷯+12﷐𝑎﷮𝑛−2﷯=3×﷐2﷮𝑛﷯
power
﷐𝑝﷮𝑛﷯=𝑐﷐2﷮𝑛﷯
﷐𝑎﷮𝑛﷯−7﷐𝑎﷮𝑛−1﷯+12﷐𝑎﷮𝑛−2﷯=3𝑛﷐2﷮𝑛﷯
Polynomial x power
﷐𝑝﷮𝑛﷯=﷐𝑐𝑛+𝑑﷯﷐2﷮𝑛﷯
﷐𝑎﷮𝑛﷯−7﷐𝑎﷮𝑛−1﷯+12﷐𝑎﷮𝑛−2﷯=2×﷐3﷮𝑛﷯
power
﷐𝑝﷮𝑛﷯=𝑐𝑛﷐3﷮𝑛﷯

﷐𝑝﷮𝑛﷯=𝑐
𝑐−7𝑐+12𝑐=30
𝑐=5

(So ﷐𝑎﷮𝑛﷯=𝐴﷐3﷮𝑛﷯+𝐵﷐4﷮𝑛﷯+5)
Ink Drawings
Ink Drawings
Ink Drawings

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