---
title: "Inverse Laplace transforms 1/(s²−3s+2)"
subject: "Mathematics III"
chapter: 3
key: "semester_3/mathematics_iii:inverse-laplace-transforms-1-s2-3s"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/inverse-laplace-transforms-1-s2-3s
site: BCT Engineering
---

# Inverse Laplace transforms 1/(s²−3s+2)

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2072 Chaitra, 2+3 marks
- Video: [Easy trick to find inverse Laplace of 2s+1/(s-1)(s-2)(s-3) by partial factorisation method #bech](https://www.youtube.com/watch?v=UzeTuAukZw0) — EM by danishwar shabir
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/inverse-laplace-transforms-1-s2-3s
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

Find the inverse Laplace transform of: i) 1/(s² − 3s + 2)  ii) 1/(s(s + 1)³).

## Short answer

i) Partial fractions: 1/(s-2)-1/(s-1) gives e^{2t}-e^t. ii) 1/s-1/(s+1)-1/(s+1)^2-1/(s+1)^3 yields 1-e^{-t}(1+t+t^2/2).

## Answer

#### i) L⁻¹[1/(s² − 3s + 2)]
s² − 3s + 2 = (s − 1)(s − 2), and 1/((s−1)(s−2)) = 1/(s − 2) − 1/(s − 1).

**Answer: e^(2t) − e^(t)**

#### ii) L⁻¹[1/(s(s + 1)³)]
Partial fractions:

```text
1/(s(s+1)³) = 1/s − 1/(s+1) − 1/(s+1)² − 1/(s+1)³
```

(Check at s = 1: 1/8 = 1 − 1/2 − 1/4 − 1/8 = 1/8 ✓.)

Use L⁻¹[1/(s+1)ⁿ] = tⁿ⁻¹ e^(−t)/(n − 1)!:

**Answer: 1 − e^(−t) − t e^(−t) − (t²/2) e^(−t) = 1 − e^(−t)(1 + t + t²/2)**

(Alternative: L⁻¹[1/(s(s+1)³)] = ∫₀ᵗ (u²/2) e^(−u) du, the same result.)

## Check yourself

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

1. What is L^-1{1/(s^2-3s+2)}?
   A. e^t - e^(2t)
   B. e^(2t) - e^t
   C. e^(2t) + e^t
   D. e^(-2t) - e^(-t)
   Answer: B. 1/((s-1)(s-2)) = 1/(s-2) - 1/(s-1). (Key point: Result for (i))

2. 1/(s(s+1)^3) decomposes into?
   A. A/s + B/(s+1)^3
   B. A/s + B/(s+1)
   C. A/s + B/(s+1) + C/(s+1)^2 + D/(s+1)^3
   D. (As+B)/(s+1)^3
   Answer: C. All powers. (Key point: Partial fraction form)

3. What is the inverse?
   A. 1 + e^(-t)(1 + t + t^2/2)
   B. 1 - e^(-t)(1 - t + t^2/2)
   C. 1 - e^(t)(1 + t + t^2/2)
   D. 1 - e^(-t)(1 + t + t^2/2)
   Answer: D. Using t^(n-1)e^(-t)/(n-1)!. (Key point: Result for (ii))

4. Dividing by s corresponds to?
   A. Integrating the inverse from 0 to t
   B. Differentiating
   C. Multiplying by t
   D. Shifting
   Answer: A. L^-1{G/s} = integral of g. (Key point: Alternative)

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