3. Below are given the Laplace transforms F(s) for different functions. Find f(t) such that L[f(t)] = F(s). You may need to use partial fractions or complete the square. (a) 1 s(s² + 4) (b) s2 + 2s + 6
3. Below are given the Laplace transforms F(s) for different functions. Find f(t) such that L[f(t)] = F(s). You may need to use partial fractions or complete the square. (a) 1 s(s² + 4) (b) s2 + 2s + 6
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem 3: Inverse Laplace Transforms**
Below are given the Laplace transforms \( F(s) \) for different functions. Find \( f(t) \) such that \( \mathcal{L}[f(t)] = F(s) \). You may need to use partial fractions or complete the square.
(a) \( \frac{1}{s(s^2 + 4)} \)
(b) \( \frac{s}{s^2 + 2s + 6} \)
---
**Explanation:**
- In problem (a), the given Laplace transform is a rational function with the quadratic polynomial in the denominator, suggesting the potential need for partial fraction decomposition.
- In problem (b), the quadratic in the denominator may require completing the square to facilitate finding the inverse Laplace transform.
These problems test the ability to apply theorems and techniques such as partial fraction decomposition and completion of the square in solving inverse Laplace transforms.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F85346d8e-a543-42db-b008-9fe5a873d3b5%2Fdb61376a-6e97-475a-bce5-23065d3ee74d%2F50bl9sg_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 3: Inverse Laplace Transforms**
Below are given the Laplace transforms \( F(s) \) for different functions. Find \( f(t) \) such that \( \mathcal{L}[f(t)] = F(s) \). You may need to use partial fractions or complete the square.
(a) \( \frac{1}{s(s^2 + 4)} \)
(b) \( \frac{s}{s^2 + 2s + 6} \)
---
**Explanation:**
- In problem (a), the given Laplace transform is a rational function with the quadratic polynomial in the denominator, suggesting the potential need for partial fraction decomposition.
- In problem (b), the quadratic in the denominator may require completing the square to facilitate finding the inverse Laplace transform.
These problems test the ability to apply theorems and techniques such as partial fraction decomposition and completion of the square in solving inverse Laplace transforms.
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