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(a) ƒ (t) = €²t sin (3t)

(a) ƒ (t) = €²t sin (3t)

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.3: The Natural Exponential Function
Problem 1E
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Question
Title: Finding the Laplace Transform of Given Functions Introduction: The Laplace Transform is a powerful integral transform used to switch a function from its original domain (time) to the frequency domain. It is extremely useful in various fields including engineering, physics, and control theory. Problem Statement: Find the Laplace Transform of each of the following functions: Functions: (a) \( f(t) = e^{2t} \sin(3t) \) (b) \( y(t) = t^2(t + 3) \) (c) \( g(t) = e^{-t} \sin^2(2t) \) Instructions: For each function, we will apply the appropriate Laplace Transform formulas and theorems. Review the properties of the Laplace Transform, including linearity, time-shifting, and frequency-shifting properties, along with standard Laplace Transforms of common functions like \(\sin(at)\), \(t^n\), etc. Detailed Explanations: 1. **For (a) \( f(t) = e^{2t} \sin(3t) \):** - We use the frequency-shifting property which states \( \mathcal{L}\{e^{at}f(t)\}(s) = F(s-a) \). - Start by finding the Laplace Transform of \(\sin(3t)\) and then apply the frequency shift. 2. **For (b) \( y(t) = t^2(t + 3) \):** - Distribute the terms to separate into \(y(t) = t^3 + 3t^2\). - Use the linearity property of the Laplace Transform to handle each term \(t^3\) and \(t^2\) separately. - Apply the standard Laplace Transform formulas for \( t^n \). 3. **For (c) \(g(t) = e^{-t} \sin^2(2t) \):** - Use the double angle identity for sine, \(\sin^2(2t) = \frac{1 - \cos(4t)}{2}\), to simplify. - Find the Laplace Transform of each term after simplification, then apply the frequency shift for \( e^{-t} \). Conclusion: By applying the rules and properties of the Laplace Transform, we can efficiently transform each of these time-domain
Transcribed Image Text:Title: Finding the Laplace Transform of Given Functions Introduction: The Laplace Transform is a powerful integral transform used to switch a function from its original domain (time) to the frequency domain. It is extremely useful in various fields including engineering, physics, and control theory. Problem Statement: Find the Laplace Transform of each of the following functions: Functions: (a) \( f(t) = e^{2t} \sin(3t) \) (b) \( y(t) = t^2(t + 3) \) (c) \( g(t) = e^{-t} \sin^2(2t) \) Instructions: For each function, we will apply the appropriate Laplace Transform formulas and theorems. Review the properties of the Laplace Transform, including linearity, time-shifting, and frequency-shifting properties, along with standard Laplace Transforms of common functions like \(\sin(at)\), \(t^n\), etc. Detailed Explanations: 1. **For (a) \( f(t) = e^{2t} \sin(3t) \):** - We use the frequency-shifting property which states \( \mathcal{L}\{e^{at}f(t)\}(s) = F(s-a) \). - Start by finding the Laplace Transform of \(\sin(3t)\) and then apply the frequency shift. 2. **For (b) \( y(t) = t^2(t + 3) \):** - Distribute the terms to separate into \(y(t) = t^3 + 3t^2\). - Use the linearity property of the Laplace Transform to handle each term \(t^3\) and \(t^2\) separately. - Apply the standard Laplace Transform formulas for \( t^n \). 3. **For (c) \(g(t) = e^{-t} \sin^2(2t) \):** - Use the double angle identity for sine, \(\sin^2(2t) = \frac{1 - \cos(4t)}{2}\), to simplify. - Find the Laplace Transform of each term after simplification, then apply the frequency shift for \( e^{-t} \). Conclusion: By applying the rules and properties of the Laplace Transform, we can efficiently transform each of these time-domain
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