Difference eq x(k+2)-4x(k+1)+4x(k)=2^k

Equation: x(k + 2) − 4x(k + 1) + 4x(k) = 2ᵏ, x(0) = 0, x(1) = 1.

Step 1: z-transform (Z[2ᵏ] = z/(z − 2)):

[z²X − z] − 4[zX] + 4X = z/(z − 2)
(z − 2)² X = z + z/(z − 2)
X(z) = z/(z − 2)² + z/(z − 2)³

Step 2: inverse transform using standard pairs

• Z[k aᵏ] = az/(z − a)², so z/(z − 2)² ↔ k·2^{k−1}
• Z[k(k − 1) aᵏ] = 2a²z/(z − a)³, so z/(z − 2)³ ↔ k(k − 1)·2ᵏ/(2·4) = k(k − 1)·2^{k−3}

x(k) = k·2^{k−1} + k(k − 1)·2^{k−3}
     = k·2^{k−3} [4 + (k − 1)]
     = k (k + 3) 2^{k−3}

Answer: x(k) = k(k + 3)·2^{k−3}, k ≥ 0.

Check: x(0) = 0, x(1) = 1, x(2) = 5 (recurrence: 4·1 − 0 + 1 = 5 ✓), x(3) = 4·5 − 4·1 + 2 = 18 ✓ (formula: 18).