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

# Difference equation by Z-transform

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2081 Chaitra (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-three
- 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) − 4x(k + 1) − 5x(k) = 0, given that x(0) = 0, x(1) = 1.

## Short answer

Z-transform yields X(z)=z/(( z-5)(z+1)). Partial fractions: X(z)=(1/6)[z/(z-5)-z/(z+1)]. Inverse gives x(k)=[5^k - (-1)^k]/6.

## Answer

x(k + 2) − 4x(k + 1) − 5x(k) = 0, x(0) = 0, x(1) = 1.

**Step 1: Z-transform.**

```text
[z²X − z·1] − 4[zX] − 5X = 0
(z² − 4z − 5) X = z
X(z) = z/((z − 5)(z + 1))
```

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

```text
X/z = 1/((z−5)(z+1)) = (1/6) [ 1/(z − 5) − 1/(z + 1) ]
X(z) = (1/6) [ z/(z − 5) − z/(z + 1) ]
```

**Step 3: Invert.**

**Answer: x(k) = [5ᵏ − (−1)ᵏ]/6**

**Check:** x(0) = 0, x(1) = (5 + 1)/6 = 1, x(2) = 4x(1) + 5x(0) = 4 and (25 − 1)/6 = 4 ✓; x(3) = 4·4 + 5·1 = 21 and (125 + 1)/6 = 21 ✓.

## Check yourself

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

1. (z^2 - 4z - 5)X = z; factor gives
   A. (z+5)(z-1)
   B. (z-5)(z-1)
   C. (z+5)(z+1)
   D. (z-5)(z+1)
   Answer: D. Roots 5 and -1. (Key point: Algebraic equation)

2. X/z =
   A. (1/6)[1/(z-5) - 1/(z+1)]
   B. (1/6)[1/(z-5) + 1/(z+1)]
   C. (1/5)[1/(z-5) - 1/(z+1)]
   D. (1/6)[1/(z+5) - 1/(z-1)]
   Answer: A. A = 1/6, B = -1/6. (Key point: Partial fractions)

3. Z^-1[z/(z-a)] =
   A. a/k
   B. a^k
   C. k^a
   D. a^(-k)
   Answer: B. Geometric sequence. (Key point: Inverse)

4. x(k) =
   A. [5^k + (-1)^k]/6
   B. [5^k - (-1)^k]/4
   C. [5^k - (-1)^k]/6
   D. [(-5)^k - 1]/6
   Answer: C. x(2) = 4 matches (25-1)/6. (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-two) — Mathematics III, chapter 3, asked 2×

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