- 2082 Chaitra (new course) · 4 marks
Difference equation by Z-transform
Short answer
Z-transform gives (z^2-4z+4)X=z, so X(z)=z/(z-2)^2. Recognize inverse as k·2^(k-1). Hence x(k)=k·2^(k-1).
Using Z-transform technique, solve the difference equation x(k + 2) − 4x(k + 1) + 4x(k) = 0 under x(0) = 0, x(1) = 1.
x(k + 2) − 4x(k + 1) + 4x(k) = 0, x(0) = 0, x(1) = 1.
Step 1: Z-transform. Use Z[x(k+1)] = zX − z x(0) and Z[x(k+2)] = z²X − z²x(0) − z x(1).
[z²X − 0 − z] − 4[zX − 0] + 4X = 0
(z² − 4z + 4) X = z
X(z) = z/(z − 2)²
Step 2: Invert. Z[k aᵏ] = a z/(z − a)². With a = 2: Z[k 2ᵏ] = 2z/(z − 2)², so z/(z − 2)² = Z[k 2ᵏ⁻¹].
Answer: x(k) = k·2^(k−1)
Check: x(0) = 0 and x(1) = 1. From the equation, x(2) = 4x(1) − 4x(0) = 4, and the formula gives 2·2¹ = 4 ✓. x(3) = 4x(2) − 4x(1) = 16 − 4 = 12, and the formula gives 3·2² = 12 ✓.
✓ Answer checked by Claude Opus 5.5 (high effort)
Related questions
- Solve ODE by Laplace transform x''+4x'+4x=6e^(−t) · Mathematics III, chapter 3 · asked 16×
- Solve ODE by Laplace transform y''−3y'+2y=4x+e^(3x) · Mathematics III, chapter 3 · asked 14×
- Difference equation by Z-transform · Mathematics III, chapter 3 · asked 2×
- Difference equation by Z-transform · Mathematics III, chapter 3 · asked 2×
- Solve ODE by Laplace transform (D²−3D+2)y=e^(3t)underth… · Mathematics III, chapter 3
- Solve ODE by Laplace transform y''+y'−2y=x · Mathematics III, chapter 3 · asked 5×
- Difference eq x(k+2)-4x(k+1)+4x(k)=2^k · Applied Mathematics, chapter 2 · asked 5×
- Difference eq x(k+2)-4x(k+1)+4x(k)=0, x(0)=1 · Applied Mathematics, chapter 2 · asked 3×
- Difference eq x(k+2)-4x(k+1)+4x(k)=0, x(0)=0 · Applied Mathematics, chapter 2 · asked 2×
- Difference equation x(k+2)+3x(k+1)+2x(k)=0 · Digital Control System, chapter 2 · asked 5×