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Which one of the following is the zero-input response of the system y[n] = 3y[n - 1] = 4y[n - 2] = 0 described by the homogeneous second-order difference equation if y[-2] = 0 and y[-1] = 5 ?

Which one of the following is the zero-input response of the system y[n] = 3y[n - 1] = 4y[n - 2] = 0 described by the homogeneous second-order difference equation if y[-2] = 0 and y[-1] = 5 ? Correct Answer y<span style="position: relative; line-height: 0; vertical-align: baseline; bottom: -0.25em; font-size: 10.5px; user-select: auto;">zi</span>(n) = (-1)<span style="position: relative; line-height: 0; vertical-align: baseline; top: -0.5em; font-size: 10.5px; user-select: auto;">n+1</span> + (4)<span style="position: relative; line-height: 0; vertical-align: baseline; top: -0.5em; font-size: 10.5px; user-select: auto;">n+2</span>, n ≥ 0

Concept:

Zero input solution:

The zero input solution is the response of the system to the initial conditions, with the input set to zero. 

The zero state solution:

The zero state solution is the response of the system to the input, with initial conditions set to zero. 

+4y(-1)
y(1)=13y(-1)+12y(-2)

From the equation (ii)
y(0)=C1+C2
and
y(1)=C1(-1)+C2(4)
y(1)=-C1+4C2

By equating these two set of relations, we have
C1+C2=3y(-1)+4y(-2)=15
-C1+4C2=13y(-1)+12y(-2)=65

On solving the above two equations we get,
C1=-1 and C2=16
Therefore the zero-input response is

Yzi(n) = (-1)n+1 + (4)n+2

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