What is the zero-input response of the system described by the homogenous second order equation y(n)-3y(n-1)-4y(n-2)=0 if the initial conditions are y(-1)=5 and y(-2)=0? - Study24x7
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Goraksha Dadhich
30 Mar 2019 11:01 AM study24x7 study24x7

What is the zero-input response of the system described by the homogenous second order equation y(n)-3y(n-1)-4y(n-2)=0 if the initial conditions are y(-1)=5 and y(-2)=0?

A

(-1)n-1 + (4)n-2

B

(-1)n+1 + (4)n+2

C

 (-1)n+1 + (4)n-2

D

None of the mentioned

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  • geet sharma
    •  Given difference equation is y(n)-3y(n-1)-4y(n-2)=0—-(1)
    Let y(n)=λn
    Substituting y(n) in the given equation
    => λn – 3λn-1 – 4λn-2 = 0
    => λn-2(λ2 – 3λ – 4) = 0
    the roots of the above equation are λ=-1,4
    Therefore, general form of the solution of the homogenous equation is

    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 y(-1) and y(-2).
    From the given equation (1)
    y(0)=3y(-1)+4y(-2)
    y(1)=3y(0)+4y(-1)
    =3[3y(-1)+4y(-2)]+4y(-1)
    =13y(-1)+12y(-2)
    From the equation (2)
    y(0)=C1+C2 and
    y(1)=C1(-1)+C2(4)=-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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    Reply 30 Mar 2019 11:49 AM

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