---
title: "Difference eq x(k+2)-4x(k+1)+4x(k)=0, x(0)=0"
subject: "Applied Mathematics"
chapter: 2
key: "semester_4/applied_mathematics:difference-eq-4x1-4x-x0-0-x1-1"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/difference-eq-4x1-4x-x0-0-x1-1
site: BCT Engineering
---

# Difference eq x(k+2)-4x(k+1)+4x(k)=0, x(0)=0

Applied Mathematics · Chapter 2: The Z-Transform

- Asked in: 2073 Bhadra, 5 marks; 2070 Magh, 5 marks (asked 2 times)
- 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-4x1-4x-x0-0-x1-1
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-2

## Question

Apply method of z-transform to solve the difference equation x(k + 2) - 4x(k + 1) + 4x(k) = 0; x(0) = 0, x(1) = 1.

## Short answer

Transform with x(0)=0, x(1)=1 yields X(z)=z/(z−2)². Invert using Z⁻¹[z/(z−a)²]=k a^(k−1): x(k)=k·2^(k−1), k≥0.

## Answer

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

**Step 1: transform.** Using Z[x(k + 2)] = z²X − z²x(0) − z x(1) and Z[x(k + 1)] = zX − z x(0):

[z²X − 0 − z(1)] − 4[zX − 0] + 4X = 0

(z² − 4z + 4) X = z

**Step 2: solve.**

X(z) = z/(z − 2)²

**Step 3: invert.** Z⁻¹[z/(z − a)²] = k aᵏ⁻¹ with a = 2:

**Answer: x(k) = k · 2ᵏ⁻¹,  k ≥ 0**

Check with x(k + 2) = 4x(k + 1) − 4x(k):

```text
k    : 0  1  2   3   4
x(k) : 0  1  4  12  32
```

The formula gives x(2) = 2 · 2 = 4, x(3) = 3 · 4 = 12, x(4) = 4 · 8 = 32. ✓

## Check yourself

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

1. What is the algebraic equation?
   A. (z²−4z+4)X = z
   B. (z²−4z+4)X = 1
   C. (z²−4z+4)X = z²
   D. (z²+4z+4)X = z
   Answer: A. Only z x(1) = z remains. (Key point: Transform)

2. Why a double pole?
   A. Roots 1 and 4
   B. z²−4z+4 = (z−2)²
   C. Roots ±2
   D. Roots 2 and 0
   Answer: B. Perfect square. (Key point: Repeated root)

3. x(k)?
   A. 2^k
   B. k·2^k
   C. k·2^{k−1}
   D. (k−1)2^k
   Answer: C. x = 0, 1, 4, 12, 32. (Key point: Result)

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