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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
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
What is the zero-input response of the system described by the homogen...
Explanation: 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
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.
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.
Most Upvoted Answer
What is the zero-input response of the system described by the homogen...
Zero-Input Response of the Homogeneous Second Order Equation
The given homogenous second order equation is y(n) - 3y(n-1) - 4y(n-2) = 0.
Initial Conditions
The initial conditions provided are y(-1) = 5 and y(-2) = 0.
Finding the Zero-Input Response
To find the zero-input response of the system, we need to solve the homogenous equation with the given initial conditions.
The characteristic equation for the given homogenous equation is λ^2 - 3λ - 4 = 0, which can be factored as (λ + 1)(λ - 4) = 0.
Therefore, the roots of the characteristic equation are λ = -1 and λ = 4.
The general solution for the homogenous equation is y(n) = A(-1)^n + B(4)^n.
Using the initial conditions y(-1) = 5 and y(-2) = 0, we can solve for the constants A and B.
Substituting the initial conditions into the general solution, we get:
5 = -A + 4B
0 = A + 16B
Solving these equations, we find A = 4 and B = 1.
Therefore, the zero-input response of the system is y(n) = (-1)^n + 4(4)^n, which simplifies to y(n) = (-1)^n + 4^n, matching with option B.
The given homogenous second order equation is y(n) - 3y(n-1) - 4y(n-2) = 0.
Initial Conditions
The initial conditions provided are y(-1) = 5 and y(-2) = 0.
Finding the Zero-Input Response
To find the zero-input response of the system, we need to solve the homogenous equation with the given initial conditions.
The characteristic equation for the given homogenous equation is λ^2 - 3λ - 4 = 0, which can be factored as (λ + 1)(λ - 4) = 0.
Therefore, the roots of the characteristic equation are λ = -1 and λ = 4.
The general solution for the homogenous equation is y(n) = A(-1)^n + B(4)^n.
Using the initial conditions y(-1) = 5 and y(-2) = 0, we can solve for the constants A and B.
Substituting the initial conditions into the general solution, we get:
5 = -A + 4B
0 = A + 16B
Solving these equations, we find A = 4 and B = 1.
Therefore, the zero-input response of the system is y(n) = (-1)^n + 4(4)^n, which simplifies to y(n) = (-1)^n + 4^n, matching with option B.
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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-2b)(-1)n+1+ (4)n+2c)(-1)n+1+ (4)n-2d)None of the mentionedCorrect answer is option 'B'. Can you explain this answer?
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