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s{3 – 9t* + 8sin(2t) + tel} 2. 7 2-¹ { / +(6+²1²5 - 11/12 + 3-2 3s s²+49 :}

s{3 – 9t* + 8sin(2t) + tel} 2. 7 2-¹ { / +(6+²1²5 - 11/12 + 3-2 3s s²+49 :}

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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# Laplace Transforms for Common Functions The table below presents the Laplace transforms for various functions, \( f(t) \). These transforms are essential tools in the analysis of linear time-invariant systems, commonly used in engineering and physics. The Laplace transform of a function \( f(t) \) is denoted by \( \mathcal{L}\{f(t)\} \) and is a function of the complex variable \( s \). | Function \( f(t) \) | \( \mathcal{L}\{f(t)\} = F(s) \) | |---|---| | 1 | \(\frac{1}{s}\) | | \( t \) | \(\frac{1}{s^2}\) | | \( t^n \) | \(\frac{n!}{s^{n+1}}, \text{ where } n \text{ is a positive integer} \) | | \( \sin (kt) \) | \(\frac{k}{s^2 + k^2}\) | | \( \cos (kt) \) | \(\frac{s}{s^2 + k^2}\) | | \( \sin^2 (kt) \) | \(\frac{2k^2}{s(s^2 + 4k^2)}\) | | \( \cos^2 (kt) \) | \(\frac{s^2 + 2k^2}{s(s^2 + 4k^2)}\) | | \( e^{at} \) | \(\frac{1}{s - a}\) | | \( t \cdot e^{at} \) | \(\frac{1}{(s - a)^2}\) | | \( t^n \cdot e^{at} \) | \(\frac{n!}{(s - a)^{n+1}}, \text{ where } n \text{ is a positive integer} \) | | \(\frac{e^{at} - e^{bt}}{a - b} \) | \(\frac{1}{(s - a)(s - b)}\) | | \(\frac{ae^{at} - be^{bt}}{a - b} \) | \(\frac{s}{(s - a)(s - b)}\) | ### Explanation of Common
Transcribed Image Text:# Laplace Transforms for Common Functions The table below presents the Laplace transforms for various functions, \( f(t) \). These transforms are essential tools in the analysis of linear time-invariant systems, commonly used in engineering and physics. The Laplace transform of a function \( f(t) \) is denoted by \( \mathcal{L}\{f(t)\} \) and is a function of the complex variable \( s \). | Function \( f(t) \) | \( \mathcal{L}\{f(t)\} = F(s) \) | |---|---| | 1 | \(\frac{1}{s}\) | | \( t \) | \(\frac{1}{s^2}\) | | \( t^n \) | \(\frac{n!}{s^{n+1}}, \text{ where } n \text{ is a positive integer} \) | | \( \sin (kt) \) | \(\frac{k}{s^2 + k^2}\) | | \( \cos (kt) \) | \(\frac{s}{s^2 + k^2}\) | | \( \sin^2 (kt) \) | \(\frac{2k^2}{s(s^2 + 4k^2)}\) | | \( \cos^2 (kt) \) | \(\frac{s^2 + 2k^2}{s(s^2 + 4k^2)}\) | | \( e^{at} \) | \(\frac{1}{s - a}\) | | \( t \cdot e^{at} \) | \(\frac{1}{(s - a)^2}\) | | \( t^n \cdot e^{at} \) | \(\frac{n!}{(s - a)^{n+1}}, \text{ where } n \text{ is a positive integer} \) | | \(\frac{e^{at} - e^{bt}}{a - b} \) | \(\frac{1}{(s - a)(s - b)}\) | | \(\frac{ae^{at} - be^{bt}}{a - b} \) | \(\frac{s}{(s - a)(s - b)}\) | ### Explanation of Common
### Laplace Transform and Inverse Laplace Transform Examples #### Problem 1: Calculating the Laplace Transform Find the Laplace Transform of the following function: \[ \mathcal{L}\{3 - 9t^2 + 8\sin(2t) + t^5 e^{3t}\} \] #### Problem 2: Calculating the Inverse Laplace Transform Find the Inverse Laplace Transform of the following expression: \[ \mathcal{L}^{-1} \left\{ \frac{9}{s^4} + \frac{7}{(s+1)^5} - \frac{11}{s-2} + \frac{3s}{s^2 + 49} \right\} \] In these problems, the symbol \(\mathcal{L}\{ \cdot \}\) denotes the Laplace Transform operator, and \(\mathcal{L}^{-1}\{ \cdot \}\) denotes the Inverse Laplace Transform operator. ### Explanation: 1. **Laplace Transform**: The problem involves finding the Laplace Transform of a time-domain function composed of polynomial, sinusoidal, and exponential terms. 2. **Inverse Laplace Transform**: This problem involves finding the time-domain function corresponding to a given Laplace domain expression. This typically requires decomposing the expression into simpler fractions that are easier to transform back. In teaching these transforms, it is valuable to recall the basic Laplace Transform pairs and properties, such as linearity, frequency shifting, and the transforms of common functions (e.g., \( e^{at} \), \( \sin(bt) \), polynomials \( t^n \)). This understanding is foundational for solving differential equations and analyzing systems in engineering and physics.
Transcribed Image Text:### Laplace Transform and Inverse Laplace Transform Examples #### Problem 1: Calculating the Laplace Transform Find the Laplace Transform of the following function: \[ \mathcal{L}\{3 - 9t^2 + 8\sin(2t) + t^5 e^{3t}\} \] #### Problem 2: Calculating the Inverse Laplace Transform Find the Inverse Laplace Transform of the following expression: \[ \mathcal{L}^{-1} \left\{ \frac{9}{s^4} + \frac{7}{(s+1)^5} - \frac{11}{s-2} + \frac{3s}{s^2 + 49} \right\} \] In these problems, the symbol \(\mathcal{L}\{ \cdot \}\) denotes the Laplace Transform operator, and \(\mathcal{L}^{-1}\{ \cdot \}\) denotes the Inverse Laplace Transform operator. ### Explanation: 1. **Laplace Transform**: The problem involves finding the Laplace Transform of a time-domain function composed of polynomial, sinusoidal, and exponential terms. 2. **Inverse Laplace Transform**: This problem involves finding the time-domain function corresponding to a given Laplace domain expression. This typically requires decomposing the expression into simpler fractions that are easier to transform back. In teaching these transforms, it is valuable to recall the basic Laplace Transform pairs and properties, such as linearity, frequency shifting, and the transforms of common functions (e.g., \( e^{at} \), \( \sin(bt) \), polynomials \( t^n \)). This understanding is foundational for solving differential equations and analyzing systems in engineering and physics.
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