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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...
Solution:
The given homogenous second-order equation is
y(n)-3y(n-1)-4y(n-2)=0
The characteristic equation of the above differential equation is
r^2-3r-4=0
Solving the above equation, we get
r=4,-1
Therefore, the general solution of the given differential equation is
y(n)= A(-1)^n + B(4)^n
where A and B are constants that can be evaluated using the initial conditions.
Given initial conditions are
y(-1)=5 and y(-2)=0
Using y(-1)=5 in the general solution, we get
5= A(-1)^(-1) + B(4)^(-1)
5= -A/1 + B/4
20= -A + B
B=20+A
Using y(-2)=0 in the general solution, we get
0= A(-1)^(-2) + B(4)^(-2)
0= -A/1 + B/16
A=16B
Substituting the value of A in the equation B=20+A, we get
B=20+16B
15B=-20
B=-4/3
Substituting the value of B in the general solution, we get
y(n)= A(-1)^n + B(4)^n
y(n)= 16B(-1)^n + B(4)^n
y(n)= (-4/3)(-1)^n + (-4/3)(4)^n
Therefore, the zero-input response of the given system is
y(n)= (-1)^n + (4)^n
The given homogenous second-order equation is
y(n)-3y(n-1)-4y(n-2)=0
The characteristic equation of the above differential equation is
r^2-3r-4=0
Solving the above equation, we get
r=4,-1
Therefore, the general solution of the given differential equation is
y(n)= A(-1)^n + B(4)^n
where A and B are constants that can be evaluated using the initial conditions.
Given initial conditions are
y(-1)=5 and y(-2)=0
Using y(-1)=5 in the general solution, we get
5= A(-1)^(-1) + B(4)^(-1)
5= -A/1 + B/4
20= -A + B
B=20+A
Using y(-2)=0 in the general solution, we get
0= A(-1)^(-2) + B(4)^(-2)
0= -A/1 + B/16
A=16B
Substituting the value of A in the equation B=20+A, we get
B=20+16B
15B=-20
B=-4/3
Substituting the value of B in the general solution, we get
y(n)= A(-1)^n + B(4)^n
y(n)= 16B(-1)^n + B(4)^n
y(n)= (-4/3)(-1)^n + (-4/3)(4)^n
Therefore, the zero-input response of the given system is
y(n)= (-1)^n + (4)^n
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What is the zero-input response of the system described by the homogen...
Find zero input response zero state response x[01-2 -1]
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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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