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f(t) = cos(2t). Find F(s)

f(t) = cos(2t). Find F(s)

Advanced Engineering Mathematics
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 1: Functions and Transformations **Objective:** Find the functions from Table 6.2.1. **a.** Given the function \( f(t) = \cos(2t) \), find \( F(s) \). **b.** For the function \( g(t) = e^{2t} \sin(3t) \), determine \( G(s) \). **c.** Consider \( H(s) = \frac{3s + 1}{s^2 - 2s + 5} \). Find the corresponding \( h(t) \). **Explanation:** - **(a)** involves the cosine function, which is commonly associated with Laplace transforms. - **(b)** combines an exponential and sine function, implying a transformation using complex exponentials. - **(c)** shows a rational function in \( s \), which typically involves partial fraction decomposition to revert to time domain. These problems focus on applying standard transformations to expressions in \( t \) and \( s \), often using Laplace transform tables.
Transcribed Image Text:## Problem 1: Functions and Transformations **Objective:** Find the functions from Table 6.2.1. **a.** Given the function \( f(t) = \cos(2t) \), find \( F(s) \). **b.** For the function \( g(t) = e^{2t} \sin(3t) \), determine \( G(s) \). **c.** Consider \( H(s) = \frac{3s + 1}{s^2 - 2s + 5} \). Find the corresponding \( h(t) \). **Explanation:** - **(a)** involves the cosine function, which is commonly associated with Laplace transforms. - **(b)** combines an exponential and sine function, implying a transformation using complex exponentials. - **(c)** shows a rational function in \( s \), which typically involves partial fraction decomposition to revert to time domain. These problems focus on applying standard transformations to expressions in \( t \) and \( s \), often using Laplace transform tables.
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