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Homogeneous solution of: y(n) -9/16y(n-2) = x(n-1)
- a)C1(3/4)n+C2(3/4)-n
- b)C1-(3/4)n-1+C2(3/4)n-1
- c)C1(3/4)n
- d)C1-(3/4)n
Correct answer is option 'A'. Can you explain this answer?
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Homogeneous solution of: y(n) -9/16y(n-2) = x(n-1)a)C1(3/4)n+C2(3/4)-n...
Explanation: Taking the z-transform of the given difference equation and solving the homogeneous equation and finding the solution using complimentary function.
Most Upvoted Answer
Homogeneous solution of: y(n) -9/16y(n-2) = x(n-1)a)C1(3/4)n+C2(3/4)-n...
To solve the given difference equation and find the homogeneous solution, we can use the method of characteristic equation.
The given difference equation is y(n) - 9/16y(n-2) = x(n-1).
1. Characteristic Equation:
To find the characteristic equation, we assume a homogeneous solution of the form y(n) = r^n, where r is a constant to be determined. Substituting this into the difference equation, we get:
r^n - 9/16r^(n-2) = 0
Simplifying the equation, we can rewrite it as:
r^n - (9/16)r^(n-2) = 0
16r^n - 9r^(n-2) = 0
16r^2 - 9 = 0
2. Solving the Characteristic Equation:
To solve the characteristic equation, we can factorize it as follows:
(4r - 3)(4r + 3) = 0
This gives us two possible values for r:
r1 = 3/4
r2 = -3/4
3. Homogeneous Solution:
The homogeneous solution of the difference equation can be written as a linear combination of the solutions obtained from the characteristic equation. Let's assume the homogeneous solution as yh(n) = C1(3/4)^n + C2(-3/4)^n, where C1 and C2 are constants to be determined.
Therefore, the correct answer is option 'A': yh(n) = C1(3/4)^n + C2(-3/4)^n.
The given difference equation is y(n) - 9/16y(n-2) = x(n-1).
1. Characteristic Equation:
To find the characteristic equation, we assume a homogeneous solution of the form y(n) = r^n, where r is a constant to be determined. Substituting this into the difference equation, we get:
r^n - 9/16r^(n-2) = 0
Simplifying the equation, we can rewrite it as:
r^n - (9/16)r^(n-2) = 0
16r^n - 9r^(n-2) = 0
16r^2 - 9 = 0
2. Solving the Characteristic Equation:
To solve the characteristic equation, we can factorize it as follows:
(4r - 3)(4r + 3) = 0
This gives us two possible values for r:
r1 = 3/4
r2 = -3/4
3. Homogeneous Solution:
The homogeneous solution of the difference equation can be written as a linear combination of the solutions obtained from the characteristic equation. Let's assume the homogeneous solution as yh(n) = C1(3/4)^n + C2(-3/4)^n, where C1 and C2 are constants to be determined.
Therefore, the correct answer is option 'A': yh(n) = C1(3/4)^n + C2(-3/4)^n.
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Homogeneous solution of: y(n) -9/16y(n-2) = x(n-1)a)C1(3/4)n+C2(3/4)-nb)C1-(3/4)n-1+C2(3/4)n-1c)C1(3/4)nd)C1-(3/4)nCorrect answer is option 'A'. Can you explain this answer?
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