Difference equation with trigonometric term Trying to find the general solution to this homogeneous difference equation: y_k - 2 cos theta y_(k-1) + y_(k-2) = 0.

Alexia Avila

Alexia Avila

Answered question

2022-11-17

Difference equation with trigonometric term
Trying to find the general solution to this homogeneous difference equation:
y k 2 cos θ y k 1 + y k 2 = 0.
The characteristic equation is
λ 2 2 cos θ λ + 1 = 0.
Not sure how to factor this, but tried
( λ cos θ ) ( λ cos θ ) = 0
but I am stuck as to how to get cos 2 θ = 1 using trigonometric identities.
By using the quadratic formula I get a discriminant of
4 ( cos 2 θ 1 )
and I am stuck on how to simplify this to get the general solution.
Any help is appreciated. This is not for homework, but self study.

Answer & Explanation

reinleikcyo

reinleikcyo

Beginner2022-11-18Added 11 answers

The logarithm function is f ( x ) = log ( x ) is defined in f : C R C
So, let z 1 = e ρ 1 + i θ 1 , z 2 = e ρ 2 + i θ 2 C , if θ 1 + θ 2 π, then
log ( z 1 z 2 ) = log ( z 1 ) + log ( z 2 ), if θ 1 + θ 2 π, then
log ( z 1 z 2 ) = log ( z 1 ) + log ( z 2 )

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