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

# Laplace transforms of functions teᵗ

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2082 Shrawan, 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-te
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

Find the Laplace transforms of (i) t eᵗ  (ii) sinh at sin at.

## Short answer

L{t eᵕ}=−d/ds(1/(s−1))=1/(s−1)². Write sinh at sin at as (e^(at)−e^(−at))/2 · sin at and apply first shifting theorem: L{sinh at sin at}=2a² s/(s⁴+4a⁴).

## Answer

#### (i) L{t eᵗ}
Use L{t f(t)} = −dF/ds with F(s) = L{eᵗ} = 1/(s − 1):

```text
L{t eᵗ} = −d/ds [1/(s − 1)] = 1/(s − 1)²      (s > 1)
```

#### (ii) L{sinh at · sin at}
Write sinh at = (e^(at) − e^(−at))/2 and use L{e^(±at) sin at} by the first shifting theorem:

```text
L{e^(at) sin at}  = a/((s − a)² + a²)
L{e^(−at) sin at} = a/((s + a)² + a²)

L{sinh at sin at} = (a/2) [ 1/(s² − 2as + 2a²) − 1/(s² + 2as + 2a²) ]
                  = (a/2) · 4as / [(s² + 2a²)² − 4a²s²]
                  = 2a² s/(s⁴ + 4a⁴)
```

since (s² + 2a²)² − 4a²s² = s⁴ + 4a⁴.

**Answers:** (i) 1/(s − 1)²; (ii) 2a²s/(s⁴ + 4a⁴).

## Check yourself

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

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

2. L{e^(at) sin at} equals what?
   A. a/((s-a)^2 + a^2)
   B. a/((s+a)^2 + a^2)
   C. s/((s-a)^2 + a^2)
   D. a/(s^2 + a^2)
   Answer: A. First shifting replaces s by s-a in a/(s^2+a^2). (Key point: Shifted sine)

3. (s^2 + 2a^2)^2 - 4a^2 s^2 simplifies to what?
   A. s^4 - 4a^4
   B. s^4 + 4a^4
   C. s^4 + 8a^4
   D. s^4 + 4a^2 s^2 + 4a^4
   Answer: B. Expanding: s^4 + 4a^2s^2 + 4a^4 - 4a^2s^2 = s^4 + 4a^4. (Key point: Key algebraic identity)

4. What is L{sinh at sin at}?
   A. 2a^2 s/(s^4 - 4a^4)
   B. a^2 s/(s^4 + 4a^4)
   C. 2a^2 s/(s^4 + 4a^4)
   D. 2a s/(s^4 + 4a^4)
   Answer: C. (a/2) * 4as/(s^4+4a^4) = 2a^2 s/(s^4 + 4a^4). (Key point: Result for sinh at sin at)

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