---
title: "Laplace transforms of functions"
subject: "Mathematics III"
chapter: 3
key: "semester_3/mathematics_iii:laplace-transforms-of-functions"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/laplace-transforms-of-functions
site: BCT Engineering
---

# Laplace transforms of functions

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2071 Chaitra, 2.5+2.5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/laplace-transforms-of-functions
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

Find the Laplace transform of the following: a) t e^(−2t) cos t  b) sinh at cos t.

## Short answer

a) t e^{-2t} cos t: ((s+2)^2-1)/(( (s+2)^2+1)^2 = (s^2+4s+3)/(s^2+4s+5)^2. b) sinh at cos t: Use exponential definition, result a(s^2-a^2-1)/(((s-a)^2+1)((s+a)^2+1).

## Answer

#### a) L[t e^(−2t) cos t]
L[cos t] = s/(s² + 1).
Multiply by t: L[t cos t] = −d/ds [s/(s²+1)] = (s² − 1)/(s² + 1)².
First shift (s → s + 2):

**Answer: ((s + 2)² − 1)/((s + 2)² + 1)² = (s² + 4s + 3)/(s² + 4s + 5)²**

#### b) L[sinh at cos t]
sinh at = (e^(at) − e^(−at))/2, so

```text
L[sinh at cos t] = ½ { L[e^(at) cos t] − L[e^(−at) cos t] }
                 = ½ [ (s − a)/((s − a)² + 1) − (s + a)/((s + a)² + 1) ]
```

Combining the fractions (numerator: (s−a)((s+a)²+1) − (s+a)((s−a)²+1) = 2a(s² − a² − 1)):

**Answer: a(s² − a² − 1) / [((s − a)² + 1)((s + a)² + 1)]**

## Check yourself

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

1. What is L{t cos t}?
   A. (1-s^2)/(s^2+1)^2
   B. (s^2-1)/(s^2+1)^2
   C. 2s/(s^2+1)^2
   D. s/(s^2+1)
   Answer: B. -d/ds s/(s^2+1). (Key point: L[t cos t])

2. What is L{t e^(-2t) cos t}?
   A. (s^2-4s+3)/(s^2-4s+5)^2
   B. (s^2+4s+3)/(s^2+1)^2
   C. (s^2+4s+3)/(s^2+4s+5)^2
   D. 2(s+2)/(s^2+4s+5)^2
   Answer: C. s -> s+2. (Key point: Result for (a))

3. L{e^(at) cos t} = ?
   A. (s+a)/((s+a)^2+1)
   B. 1/((s-a)^2+1)
   C. s/((s-a)^2+1)
   D. (s-a)/((s-a)^2+1)
   Answer: D. First shifting. (Key point: Shifted cosine)

4. What is L{sinh at cos t}?
   A. a(s^2-a^2-1)/[((s-a)^2+1)((s+a)^2+1)]
   B. 2a(s^2-a^2-1)/[((s-a)^2+1)((s+a)^2+1)]
   C. a(s^2+a^2+1)/[((s-a)^2+1)((s+a)^2+1)]
   D. as/[((s-a)^2+1)((s+a)^2+1)]
   Answer: A. (1/2)*2a(...) . (Key point: Result for (b))

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