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

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

Applied Mathematics · Chapter 2: The Z-Transform

- Asked in: 2073 Magh, 5 marks; 2069 Bhadra; 2070 Bhadra, 5 marks (asked 3 times)
- 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-2x1-3x-x0-0-x1-2
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-2

## Question

Solve by using z-transform the difference equation x(k + 2) + 2x(k + 1) + 3x(k) = 0 given that x(0) = 0 and x(1) = 2.

## Short answer

Taking z-transform yields X(z)=2z/(z^2+2z+3). Inverting gives x(k)=√2 (√3)^k sin(kθ) where θ=cos^{-1}(-1/√3).

## Answer

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

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

[z²X − z²(0) − z(2)] + 2[zX − z(0)] + 3X = 0

(z² + 2z + 3) X(z) = 2z

**Step 2: solve for X(z):**

X(z) = 2z/(z² + 2z + 3)

**Step 3: invert.** Roots of z² + 2z + 3: z = −1 ± i√2, modulus r = √3, and r cos θ = −1, r sin θ = √2, so cos θ = −1/√3, θ = cos⁻¹(−1/√3) ≈ 125.26° (2.186 rad).

Write the denominator as z² − 2r cos θ z + r². Using Z[rᵏ sin kθ] = r sin θ · z/(z² − 2r cos θ z + r²), with r sin θ = √2:

X(z) = (2/√2) · √2 z/(z² + 2z + 3) = √2 · Z[(√3)ᵏ sin kθ]

**Answer:**

x(k) = √2 (√3)ᵏ sin(kθ),  θ = cos⁻¹(−1/√3),  k = 0, 1, 2, …

**Check** with the recurrence x(k + 2) = −2x(k + 1) − 3x(k):

```text
k : 0   1   2   3   4    5
x : 0   2  -4   2   8  -22
```

The formula gives x(1) = √2·√3·√(2/3) = 2, x(2) = −4, x(3) = 2, matching the recurrence.

## Check yourself

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

1. What is Z[x(k+2)]?
   A. z²X - x(0) - x(1)
   B. z²X
   C. z²X - z x(0)
   D. z²X - z²x(0) - z x(1)
   Answer: D. Shifting theorem. (Key point: Transform of x(k+2))

2. What does the equation become?
   A. (z²+2z+3)X = 2z
   B. (z²+2z+3)X = 2
   C. (z²-2z+3)X = 2z
   D. (z²+2z+3)X = 0
   Answer: A. Only z·x(1) = 2z remains. (Key point: Algebraic equation)

3. What is the modulus of the roots of z²+2z+3?
   A. 3
   B. √3
   C. 1
   D. √2
   Answer: B. Product of roots 3, complex conjugates. (Key point: Roots)

4. What is x(k)?
   A. (√3)^k cos(kθ)
   B. √2 (√3)^k cos(kθ)
   C. √2 (√3)^k sin(kθ), θ = cos⁻¹(-1/√3)
   D. 2^k sin kθ
   Answer: C. Gives 0, 2, -4, 2, ... (Key point: Solution)

## Close variants set in other exams

- [Difference eq x(k+2)-4x(k+1)+4x(k)=2^k](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Adifference-eq-forced-2k-4) — Applied Mathematics, chapter 2, asked 5×
- [Difference eq x(k+2)+6x(k+1)+9x(k)=2^k](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Adifference-eq-forced-2k-9) — Applied Mathematics, chapter 2, asked 5×

## Related in Applied Mathematics

- [Difference eq x(k+2)+3x(k+1)+2x(k)=0](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Adifference-eq-3x1-2x-x0-0-x1-1) — Applied Mathematics, chapter 2, asked 3×
- [Difference eq x(k+2)-3x(k+1)+2x(k)=4^k](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Adifference-eq-forced-4k) — Applied Mathematics, chapter 2
- [Difference eq x(k+2)-4x(k+1)+4x(k)=0, x(0)=1](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Adifference-eq-4x1-4x-x0-1-x1-0) — Applied Mathematics, chapter 2, asked 3×
- [Difference eq x(k+2)-4x(k+1)+4x(k)=0, x(0)=0](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Adifference-eq-4x1-4x-x0-0-x1-1) — Applied Mathematics, chapter 2, asked 2×
- [Difference eq y(n+2)+2y(n+1)+y(n)=n](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Adifference-eq-forced-n) — Applied Mathematics, chapter 2
- [Difference eq with unit step input](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Adifference-eq-unit-step) — Applied Mathematics, chapter 2

## Related in other subjects

- [Difference equation x(k+2)+3x(k+1)+2x(k)=0](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Adifference-equation-x0-0-x1-1) — Digital Control System, chapter 2, asked 5×
- [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×
- [Solve ODE by Laplace transform y''−3y'+2y=4x+e^(3x)](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-y-ten) — Mathematics III, chapter 3, asked 14×
- [Solve ODE by Laplace transform x''+2x'+5x=e^(−t)sint](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-x) — Mathematics III, chapter 3, asked 17×