---
title: "Difference equation by Z-transform"
subject: "Mathematics III"
chapter: 3
key: "semester_3/mathematics_iii:difference-equation-by-z-transform-two"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/difference-equation-by-z-transform-two
site: BCT Engineering
---

# Difference equation by Z-transform

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2082 Kartik (new course), 4 marks
- Video: [Applications To Difference Equations Z Transforms -Problem 8](https://www.youtube.com/watch?v=bk_TBQhxLv8) — Btech Maths Hub
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/difference-equation-by-z-transform-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

Apply Z-transform to solve the difference equation x(k + 2) − x(k + 1) − 2x(k) = 0, given that x(0) = 0, x(1) = 1.

## Short answer

After Z-transform, X(z)=z/(( z-2)(z+1)). Partial fractions give X(z)=(1/3)[z/(z-2)-z/(z+1)]. Inverse yields x(k)=[2^k - (-1)^k]/3.

## Answer

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

**Step 1: Z-transform.**

```text
[z²X − z²·0 − z·1] − [zX − z·0] − 2X = 0
(z² − z − 2) X = z
X(z) = z/((z − 2)(z + 1))
```

**Step 2: Partial fractions on X/z.**

```text
X/z = 1/((z−2)(z+1)) = (1/3) [ 1/(z − 2) − 1/(z + 1) ]
X(z) = (1/3) [ z/(z − 2) − z/(z + 1) ]
```

**Step 3: Invert** using Z⁻¹[z/(z − a)] = aᵏ.

**Answer: x(k) = [2ᵏ − (−1)ᵏ]/3**

**Check:** x(0) = 0, x(1) = (2 + 1)/3 = 1, x(2) = x(1) + 2x(0) = 1 and (4 − 1)/3 = 1 ✓; x(3) = x(2) + 2x(1) = 3 and (8 + 1)/3 = 3 ✓.

## Check yourself

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

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

2. X/z = 1/((z-2)(z+1)) equals
   A. (1/3)[1/(z-2) + 1/(z+1)]
   B. [1/(z-2) - 1/(z+1)]
   C. (1/3)[1/(z+1) - 1/(z-2)]
   D. (1/3)[1/(z-2) - 1/(z+1)]
   Answer: D. A = 1/3 at z = 2, B = -1/3 at z = -1. (Key point: Partial fractions)

3. x(k) =
   A. [2^k - (-1)^k]/3
   B. [2^k + (-1)^k]/3
   C. [2^k - (-1)^k]/2
   D. [(-2)^k - 1]/3
   Answer: A. x(0) = 0, x(1) = 1. (Key point: Solution)

## Close variants set in other exams

- [Difference equation by Z-transform](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Adifference-equation-by-z-transform-three) — Mathematics III, chapter 3, asked 2×

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