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

# Difference equation by Z-transform

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2082 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
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

Using Z-transform technique, solve the difference equation x(k + 2) − 4x(k + 1) + 4x(k) = 0 under x(0) = 0, x(1) = 1.

## Short answer

Z-transform gives (z^2-4z+4)X=z, so X(z)=z/(z-2)^2. Recognize inverse as k·2^(k-1). Hence x(k)=k·2^(k-1).

## Answer

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

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

```text
[z²X − 0 − z] − 4[zX − 0] + 4X = 0
(z² − 4z + 4) X = z
X(z) = z/(z − 2)²
```

**Step 2: Invert.** Z[k aᵏ] = a z/(z − a)². With a = 2: Z[k 2ᵏ] = 2z/(z − 2)², so z/(z − 2)² = Z[k 2ᵏ⁻¹].

**Answer: x(k) = k·2^(k−1)**

**Check:** x(0) = 0 and x(1) = 1. From the equation, x(2) = 4x(1) − 4x(0) = 4, and the formula gives 2·2¹ = 4 ✓. x(3) = 4x(2) − 4x(1) = 16 − 4 = 12, and the formula gives 3·2² = 12 ✓.

## Check yourself

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

1. Z[x(k+2)] =
   A. z^2 X - z^2 x(0) - z x(1)
   B. z^2 X - z x(0)
   C. z^2 X + z x(1)
   D. z^2 X
   Answer: A. Remove the first two terms and multiply. (Key point: Shift formula)

2. With x(0)=0, x(1)=1, the equation gives X(z) =
   A. z/(z+2)^2
   B. z/(z-2)^2
   C. 1/(z-2)^2
   D. z/(z^2 - 4)
   Answer: B. (z^2 - 4z + 4)X = z. (Key point: Algebraic equation)

3. Z[k a^k] =
   A. z/(z-a)^2
   B. a z/(z-a)
   C. a z/(z-a)^2
   D. z/(z-a)^2 a^k
   Answer: C. Hence z/(z-2)^2 = Z[k 2^(k-1)]. (Key point: Inverse of repeated pole)

4. x(k) =
   A. k 2^k
   B. 2^k - 1
   C. (k-1) 2^k
   D. k 2^(k-1)
   Answer: D. x(1) = 1, x(2) = 4, x(3) = 12. (Key point: Solution)

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