---
title: "Difference eq x(k+2)-4x(k+1)+4x(k)=2^k"
subject: "Applied Mathematics"
chapter: 2
key: "semester_4/applied_mathematics:difference-eq-forced-2k-4"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/difference-eq-forced-2k-4
site: BCT Engineering
---

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

Applied Mathematics · Chapter 2: The Z-Transform

- Asked in: 2071 Magh, 5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/difference-eq-forced-2k-4
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-2

## Question

Using z-transform solve the difference equation x(k + 2) - 4x(k + 1) + 4x(k) = 2ᵏ given that x(0) = 0, x(1) = 1.

## Short answer

Z-transform yields X(z)=z/(z-2)^2+z/(z-2)^3. Inverse using standard pairs gives x(k)=k(k+3)2^(k-3).

## Answer

**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)):

```text
[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}

```text
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).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. (z-2)^2 X =
   A. z + z/(z-2)
   B. z/(z-2)
   C. z - z/(z-2)
   D. z^2 + z/(z-2)
   Answer: A. Initial x(1)=1 gives z. (Key point: Transform)

2. X(z) =
   A. z/(z-2) + z/(z-2)^2
   B. z/(z-2)^2 + z/(z-2)^3
   C. z/(z-2)^3
   D. z^2/(z-2)^2
   Answer: B. Divide each term. (Key point: Split)

3. z/(z-2)^2 inverts to
   A. k 2^k
   B. 2^k
   C. k 2^(k-1)
   D. k^2 2^k
   Answer: C. Z[k a^k]=az/(z-a)^2. (Key point: Pair)

4. x(k) =
   A. k(k+1) 2^k
   B. (k+3) 2^(k-3)
   C. k(k-3) 2^(k-3)
   D. k(k+3) 2^(k-3)
   Answer: D. x(2)=5, x(3)=18. (Key point: Result)

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