---
title: "Difference eq x(k+2)+3x(k+1)+2x(k)=0"
subject: "Applied Mathematics"
chapter: 2
key: "semester_4/applied_mathematics:difference-eq-3x1-2x-x0-0-x1-1"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/difference-eq-3x1-2x-x0-0-x1-1
site: BCT Engineering
---

# Difference eq x(k+2)+3x(k+1)+2x(k)=0

Applied Mathematics · Chapter 2: The Z-Transform

- Asked in: 2075 Baisakh, 5 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_4/applied_mathematics/difference-eq-3x1-2x-x0-0-x1-1
- 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) + 3x(k + 1) + 2x(k) = 0 with conditions x(0) = 0, x(1) = 1.

## Short answer

Take z-transform with initial conditions to get X(z)=z/[(z+1)(z+2)], partial fractions give z/(z+1)-z/(z+2). Inverse gives x(k)=(-1)^k-(-2)^k for k≥0.

## Answer

**Equation:** x(k + 2) + 3x(k + 1) + 2x(k) = 0, x(0) = 0, x(1) = 1.

**Step 1: take the z-transform** using

- Z[x(k+1)] = zX(z) − z x(0)
- Z[x(k+2)] = z²X(z) − z² x(0) − z x(1)

```text
[z²X − z²·0 − z·1] + 3[zX − z·0] + 2X = 0
(z² + 3z + 2) X = z
X(z) = z / [(z + 1)(z + 2)]
```

**Step 2: partial fractions.**

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

X(z) = z/(z + 1) − z/(z + 2)

**Step 3: inverse transform.**

**Answer:** x(k) = (−1)ᵏ − (−2)ᵏ, k ≥ 0.

Check with the recurrence x(k+2) = −3x(k+1) − 2x(k): x(2) = −3, x(3) = 7, x(4) = −15. The formula gives −3, 7, −15 ✓.

## 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
   C. z^2 X - z x(0)
   D. z^2 X + z^2 x(0)
   Answer: A. Initial values enter. (Key point: Shift rule)

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

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

4. x(k) =
   A. (-2)^k - (-1)^k
   B. 1 - 2^k
   C. (-1)^k + (-2)^k
   D. (-1)^k - (-2)^k
   Answer: D. x(2) = -3 matches recurrence. (Key point: Result)

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