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 ?

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. 

Important Points The complete response is simply the sum of the zero input and zero state response.

Solution:

Given difference equation is

y(n)-3y(n-1)-4y(n-2)=0     ...(i)
Let y(n)=λn
Substituting y(n) in equation (i)
λn – 3λn-1 – 4λn-2 = 0
λn-2(λ2 – 3λ – 4) = 0
the roots of the above equation are λ=-1,4

Therefore, the general form of the solution of the homogenous equation is
yh(n)=C1 λ1n+C2 λ2n
=C1(-1)n+C2(4)n       ...(ii)

The zero-input response of the system can be calculated from the homogenous solution by evaluating the constants in the above equation.
given the initial conditions are:

y(-1)=5 and y(-2)=0.

From the given equation (i)
y(0)=3y(-1)+4y(-2)
and
y(1)=3y(0)+4y(-1)
y(1)=3[3y(-1)+4y(-2)]+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