---
title: "Difference eq x(k+2)-3x(k+1)+2x(k)=4^k"
subject: "Applied Mathematics"
chapter: 2
key: "semester_4/applied_mathematics:difference-eq-forced-4k"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/difference-eq-forced-4k
site: BCT Engineering
---

# Difference eq x(k+2)-3x(k+1)+2x(k)=4^k

Applied Mathematics · Chapter 2: The Z-Transform

- Asked in: 2072 Asoj, 5 marks
- 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-forced-4k
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-2

## Question

Solve difference equation x(k + 2) - 3x(k + 1) + 2x(k) = 4ᵏ for x(0) = 0 and x(1) = 1.

## Short answer

Take z-transform, obtain X(z)=z(z-3)/[(z-1)(z-2)(z-4)]. Partial fractions give x(k)=2^(k-1)+4^k/6-2/3.

## Answer

**Equation:** x(k + 2) − 3x(k + 1) + 2x(k) = 4ᵏ, x(0) = 0, x(1) = 1.

**Step 1: z-transform** (Z[4ᵏ] = z/(z − 4)):

```text
[z²X − z] − 3[zX] + 2X = z/(z − 4)
(z² − 3z + 2) X = z + z/(z − 4)
(z − 1)(z − 2) X = z (z − 3)/(z − 4)
X(z) = z (z − 3) / [(z − 1)(z − 2)(z − 4)]
```

**Step 2: partial fractions.**

X(z)/z = (z − 3)/[(z − 1)(z − 2)(z − 4)] = A/(z − 1) + B/(z − 2) + C/(z − 4)

```text
A = (1 − 3)/[(1 − 2)(1 − 4)] = (−2)/(3)  = −2/3
B = (2 − 3)/[(2 − 1)(2 − 4)] = (−1)/(−2) = 1/2
C = (4 − 3)/[(4 − 1)(4 − 2)] = 1/6
```

X(z) = −(2/3) z/(z − 1) + ½ z/(z − 2) + (1/6) z/(z − 4)

**Step 3: inverse.**

**Answer:** x(k) = ½·2ᵏ + (1/6)·4ᵏ − 2/3 = 2^{k−1} + 4ᵏ/6 − 2/3.

Check: x(0) = 0 ✓, x(1) = 1 ✓, x(2) = 4 (recurrence: 3·1 − 0 + 1 = 4 ✓), x(3) = 3·4 − 2·1 + 4 = 14 ✓ (formula: 4 + 64/6 − 2/3 = 14).

## Check yourself

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

1. Z[4^k] =
   A. z/(z-4)
   B. z/(z+4)
   C. 4z/(z-1)
   D. 1/(z-4)
   Answer: A. Geometric. (Key point: Forcing transform)

2. X(z) =
   A. z/[(z-1)(z-2)(z-4)]
   B. z(z-3)/[(z-1)(z-2)(z-4)]
   C. z(z+3)/[(z-1)(z-2)(z-4)]
   D. (z-3)/[(z-1)(z-2)]
   Answer: B. z + z/(z-4) = z(z-3)/(z-4). (Key point: Algebra)

3. A, B, C are
   A. 2/3, 1/2, 1/6
   B. -2/3, -1/2, 1/6
   C. -2/3, 1/2, 1/6
   D. 1/3, 1/2, 1/6
   Answer: C. Cover-up. (Key point: Coefficients)

4. x(k) =
   A. 2^k + 4^k - 1
   B. 4^k/6 - 2^(k-1) + 2/3
   C. 2^(k-1) + 4^k/6 + 2/3
   D. 2^(k-1) + 4^k/6 - 2/3
   Answer: D. x(0)=0 and x(1)=1 check. (Key point: Result)

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