---
title: "Laplace transforms of functions t²cosat"
subject: "Mathematics III"
chapter: 3
key: "semester_3/mathematics_iii:laplace-transforms-of-functions-t2cosat"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/laplace-transforms-of-functions-t2cosat
site: BCT Engineering
---

# Laplace transforms of functions t²cosat

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2076 Asoj, 2+3 marks
- Video: [Problem 1 Based on Multiplication By 't' Property - Laplace Transform - Engineering Mathematics 3](https://www.youtube.com/watch?v=B_f5rJ-Bues) — Ekeeda
- 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-t2cosat
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

Find the Laplace transform of: i) t² cos at  ii) (1 − cosh at)/t.

## Short answer

i) Using L{t² f(t)} = d²F/ds² with F(s)=s/(s²+a²) yields 2s(s²−3a²)/(s²+a²)³. ii) Using L{f(t)/t} = ∫ₛ^∞ F(u) du with F(s)=1/s − s/(s²−a²) gives ½ ln((s²−a²)/s²).

## Answer

#### i) L{t² cos at}
Use L{t² f(t)} = d²F/ds² with F(s) = s/(s² + a²).

```text
F'(s)  = (a² − s²)/(s² + a²)²
F''(s) = [−2s(s² + a²)² − (a² − s²)·2(s² + a²)·2s]/(s² + a²)⁴
       = [−2s(s² + a²) − 4s(a² − s²)]/(s² + a²)³
       = (2s³ − 6a²s)/(s² + a²)³
       = 2s(s² − 3a²)/(s² + a²)³
```

#### ii) L{(1 − cosh at)/t}
Use L{f(t)/t} = ∫ₛ^∞ F(u) du with F(s) = 1/s − s/(s² − a²).

```text
∫ₛ^∞ [1/u − u/(u² − a²)] du = [ln u − ½ ln(u² − a²)]ₛ^∞
 = 0 − ln(s/√(s² − a²))
 = ln(√(s² − a²)/s) = ½ ln((s² − a²)/s²)        (s > |a|)
```

**Answers:** i) 2s(s² − 3a²)/(s² + a²)³; ii) ½ ln((s² − a²)/s²).

## Check yourself

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

1. L{t^2 f(t)} equals what?
   A. -d^2F/ds^2
   B. dF/ds
   C. integral of F from s to infinity
   D. d^2F/ds^2
   Answer: D. (-1)^2 = 1. (Key point: Formula for t^2)

2. For F = s/(s^2+a^2), F' = ?
   A. (a^2 - s^2)/(s^2+a^2)^2
   B. (s^2 - a^2)/(s^2+a^2)^2
   C. (s^2 + a^2)/(s^2 + a^2)^2
   D. 2s/(s^2+a^2)^2
   Answer: A. Quotient rule. (Key point: First derivative)

3. What is L{t^2 cos at}?
   A. 2s(3a^2 - s^2)/(s^2 + a^2)^3
   B. 2s(s^2 - 3a^2)/(s^2 + a^2)^3
   C. 2s(s^2 + 3a^2)/(s^2 + a^2)^3
   D. 2s/(s^2 + a^2)^2
   Answer: B. F'' = (2s^3 - 6a^2 s)/(s^2+a^2)^3. (Key point: Result for (i))

4. What is L{(1 - cosh at)/t}?
   A. (1/2) ln(s^2/(s^2 - a^2))
   B. ln((s^2 - a^2)/s^2)
   C. (1/2) ln((s^2 - a^2)/s^2)
   D. (1/2) ln((s^2 + a^2)/s^2)
   Answer: C. The integral is 0 - ln(s/sqrt(s^2 - a^2)). (Key point: Result for (ii))

## Close variants set in other exams

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