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

# Laplace transforms of functions teᵗ

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2081 Baishakh, 1+2+2 marks
- Video: [Intro to the Laplace Transform & Three Examples](https://www.youtube.com/watch?v=KqokoYr_h1A) — Dr. Trefor Bazett
- 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-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

Discuss the existence of the Laplace transform. Find the Laplace transform of (i) t eᵗ  (ii) sinh at sin at.

## Short answer

Existence: f piecewise continuous and of exponential order. (i) 1/(s-1)^2; (ii) 2 a^2 s/(s^4 + 4 a^4).

## Answer

**Existence condition.** The Laplace transform L{f(t)} = ∫₀^∞ e^(−st) f(t) dt exists for s > k if f(t) is piecewise continuous on every finite interval [0, T] and of exponential order, that is |f(t)| ≤ M e^(kt) for all t ≥ T. (These conditions are sufficient, not necessary.)

#### (i) L{t eᵗ}
```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 the first shifting theorem:

```text
L{sinh at sin at} = (1/2) [ a/((s − a)² + a²) − a/((s + a)² + a²) ]
                  = (a/2) · [(s + a)² + a² − (s − a)² − a²] / [(s² + 2a²)² − 4a²s²]
                  = (a/2) · 4as/(s⁴ + 4a⁴)
                  = 2a² s/(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. Which conditions guarantee that L{f(t)} exists for s > k?
   A. f is differentiable everywhere and bounded
   B. f is piecewise continuous on every finite interval and of exponential order |f(t)| <= M e^(kt)
   C. f is periodic and continuous at t = 0
   D. f is continuous with f(0) = 0 and f(t) -> 0 as t -> infinity
   Answer: B. The answer states these two conditions: piecewise continuity on each [0, T] and exponential order. They are sufficient, and the transform then exists for s > k. (Key point: Conditions for the Laplace transform to exist)

2. What is the logical status of the existence conditions (piecewise continuous + exponential order)?
   A. Necessary and sufficient
   B. Necessary but not sufficient
   C. Sufficient but not necessary
   D. Neither necessary nor sufficient
   Answer: C. The answer notes the conditions are sufficient, not necessary: a function can have a transform without meeting them. (Key point: Sufficient, not necessary)

3. 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. (Key point: Part (i))

4. [(s+a)^2 + a^2] - [(s-a)^2 + a^2] equals what?
   A. 4as
   B. 2as
   C. 4a^2
   D. 2a^2
   Answer: A. (s+a)^2 - (s-a)^2 = 4as. (Key point: Numerator simplification)

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