**Consider** $ F(s) = \frac{1}{s(s+a)} $**Invert and find** $ f(t) $ **by**a. Direct inspection b. Residues c. Convolution### Explanation:This text presents a mathematical problem involving the inversion of the Laplace Transform. Given the function $ F(s) = \frac{1}{s(s+a)} $, the task is to find the corresponding time-domain function $ f(t) $ using three different methods:1. **Direct Inspection** - This method involves recognizing the form of the Laplace transform and matching it with known transforms to directly find $ f(t) $.2. **Residues** - This method involves using the Residue Theorem from complex analysis to evaluate the inverse Laplace transform by finding the residues of the poles of the function in the complex plane.3. **Convolution** - This approach utilizes the convolution theorem, which states that the inverse Laplace transform of a product of two functions is the convolution of their respective inverse transforms.This educational prompt provides a practical exercise in applying different mathematical techniques to solve inverse Laplace transformations.

F(s)=1s1s+aBy inspection,f(t)= 1a-e-ata

Answered: Consider F(s) =- S s+a Invert and find…

Math

Advanced Math

Consider F(s) =- S s+a Invert and find f(t) by a. Direct inspection b. Residues c. Convolution

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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Question

**Consider** $ F(s) = \frac{1}{s(s+a)} $

**Invert and find** $ f(t) $ **by**

a. Direct inspection
b. Residues
c. Convolution

### Explanation:
This text presents a mathematical problem involving the inversion of the Laplace Transform. Given the function $ F(s) = \frac{1}{s(s+a)} $, the task is to find the corresponding time-domain function $ f(t) $ using three different methods:

1. **Direct Inspection** - This method involves recognizing the form of the Laplace transform and matching it with known transforms to directly find $ f(t) $.

2. **Residues** - This method involves using the Residue Theorem from complex analysis to evaluate the inverse Laplace transform by finding the residues of the poles of the function in the complex plane.

3. **Convolution** - This approach utilizes the convolution theorem, which states that the inverse Laplace transform of a product of two functions is the convolution of their respective inverse transforms.

This educational prompt provides a practical exercise in applying different mathematical techniques to solve inverse Laplace transformations.

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