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

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

Applied Mathematics · Chapter 2: The Z-Transform

- Asked in: 2074 Bhadra, 5 marks; 2068 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-x0-1-x1-0
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-2

## Question

Solve the difference equation by the application of z-transform: x(k + 2) - 4x(k + 1) + 4x(k) = 0 with conditions x(0) = 1, x(1) = 0.

## Short answer

Applying z-transform with conditions gives X(z)=z(z−4)/(z−2)². Partial fractions: X(z)=z/(z−2)−2z/(z−2)². Inversion yields x(k)=(1−k)·2^k, k≥0.

## Answer

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

**Step 1: transform.**

[z²X − z²x(0) − z x(1)] − 4[zX − z x(0)] + 4X = 0

(z² − 4z + 4) X = z² x(0) + z x(1) − 4z x(0) = z² − 4z

**Step 2: solve.**

X(z) = z (z − 4)/(z − 2)²

**Step 3: partial fractions on X/z.**

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

X(z) = z/(z − 2) − 2z/(z − 2)²

**Step 4: invert** using Z⁻¹[z/(z − a)] = aᵏ and Z⁻¹[z/(z − a)²] = k aᵏ⁻¹:

x(k) = 2ᵏ − 2 · k 2ᵏ⁻¹ = 2ᵏ − k 2ᵏ

**Answer: x(k) = (1 − k) 2ᵏ,  k ≥ 0**

Check from the recurrence x(k + 2) = 4x(k + 1) − 4x(k):

```text
k    : 0   1   2    3    4
x(k) : 1   0  -4  -16  -48
```

The formula gives x(2) = (−1)(4) = −4, x(3) = (−2)(8) = −16. ✓

## Check yourself

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

1. RHS after transforming?
   A. z² − 4z
   B. z²
   C. z² + 4z
   D. −4z
   Answer: A. z²x(0) + z x(1) − 4z x(0). (Key point: Transform)

2. What is X(z)?
   A. z/(z−2)²
   B. z(z−4)/(z−2)²
   C. z(z+4)/(z−2)²
   D. (z−4)/(z−2)
   Answer: B. Divide by (z−2)². (Key point: X(z))

3. Which pair is used?
   A. Z⁻¹[z/(z−a)²] = a^k
   B. Z⁻¹[z/(z−a)²] = k a^k
   C. Z⁻¹[z/(z−a)²] = k a^{k−1}
   D. Z⁻¹[z/(z−a)²] = (k−1) a^k
   Answer: C. Standard. (Key point: Inverse)

4. x(k)?
   A. (1+k)2^k
   B. 2^k − k
   C. (k−1)2^k
   D. (1−k)2^k
   Answer: D. x(2) = −4. (Key point: Result)

## Close variants set in other exams

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