Difference eq with unit step input

Equation: x(k + 2) − x(k + 1) + 0.25x(k) = u(k), x(0) = 1, x(1) = 2, with u(k) = 1 for k ≥ 0, so Z[u(k)] = z/(z − 1).

Step 1: z-transform.

[z²X − z² − 2z] − [zX − z] + 0.25X = z/(z − 1)
(z² − z + 0.25) X = z² + z + z/(z − 1)
(z − ½)² X = z³ / (z − 1)           (since (z²+z)(z−1) + z = z³)
X(z) = z³ / [(z − 1)(z − ½)²]

Step 2: partial fractions.

X/z = z² / [(z − 1)(z − ½)²] = A/(z − 1) + B/(z − ½) + C/(z − ½)²

A = 1²/(1 − ½)²                    = 4
C = (½)²/(½ − 1)                   = −1/2
B: d/dz [z²/(z−1)] at z = ½
   = (z² − 2z)/(z − 1)² = (¼ − 1)/¼ = −3
(check: A + B = 1, the z² coefficient ✓)

X(z) = 4 z/(z − 1) − 3 z/(z − ½) − ½ z/(z − ½)²

Step 3: inverse. Z⁻¹[z/(z − a)²] = k a^{k−1}, so −½ · k (½)^{k−1} = −k (½)ᵏ.

x(k) = 4 − 3(½)ᵏ − k(½)ᵏ

Answer: x(k) = 4 − (k + 3)(½)ᵏ, k ≥ 0.

Check: x(0) = 4 − 3 = 1 ✓, x(1) = 4 − 4/2 = 2 ✓, x(2) = 4 − 5/4 = 2.75 (recurrence: 2 − 0.25 + 1 = 2.75 ✓), x(3) = 3.25 ✓. As k → ∞, x(k) → 4 (steady state).