Before using the table expression must be in most basic form # Laplace Transforms for Common FunctionsThe 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 TransformFind 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 TransformFind 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.

1. L3-9t2+8sin2t+t5e3t 2. L-19s4+7s+15-11s-2+3ss2+49

Answered: s{3 – 9t* + 8sin(2t) + tel} 2. 7 2-¹ {…

Math

Advanced Math

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

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: how yo write 24/40 in its simplist way

Q: Part 2: How can the following expressions be simplified? Expression 1: Expression 2: Expression 3:…

Q: Consider the following postfix representation of an expression composed of letters and the addition,…

A: Given: Postfix expression: AB+CD-E*F+* To find: Infix expression of the given postfix expression.

Q: How to write a algebraic rule?

Q: Simplify 3 3 1 1 Enter an algebraic expre a2 ь2 Preview Syntax eriTor Enter your answer as an exp Be…

A: Given,

Q: Simplify the following expression as far as possible: (x - 6)(x+ 6) – (x+ 6)² – (x – 6)². Note that…

A: Arithmetic expression

Q: Single term algebraic expression is called?

A: Single term algebraic expression is called monomial.

Q: Does the algebraic rule match the situation? yes/no explaination: y=2x-1

A: An algebraic rule is a mathematical expression that relates two variables and is written in the form…

Q: A stone is thrown into the air. The height based on time, t, can be modeled by the expression below.…

A: NOTE: Refresh your page if you can't see any equations. . to find the starting position of the…

Q: use order of operations and simplify (2/9)2*(9/13)

A: To simplify the expression.

Q: Evaluate

A: We will factor 36 first 36=2x2x3x3 Then use it to simplify Used sqrt(-1)=i Answer(a): 6i

Q: What does the following expression represent? Do not perform any calculations; rather just write out…

Question

Before using the table expression must be in most basic form

# 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.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

part-1

VIEW

part-2

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 3 steps

SEE SOLUTION Check out a sample Q&A here

Similar questions

I need to know how to evaluate an algebra expression on the TI 30xs calculator
Differerantiate the following expression (2X^2+4X-6)/X^2-3
Pls show complete steps, show using excel. And explanation is mandatory pls.
Draw a picture for the expression. Write a new equivalent expression without brackets. a) 2(x + 4) AO000 AO000 b) 3(2x + 3) c) 2(3x+1) d) 2(3x + 3) 2x + 8 Ee 2(4x+ 3) f) 3(x + 5) g) 5(2x + 3) h) 4(3x + 1)