What is the solution to the inverse Laplace transform of a given integral?

In summary, the student attempted to solve a multiple choice question by using the inverse Laplace transform and the convolution integral. They initially got an answer of 3 H(t-3) sin(9(t-3)) but it did not match any of the given options. They then eliminated options B, C, and F, leaving options A, D, and E. They tried using the convolution integral, but it yielded a different answer. Eventually, they realized that the correct answer was E, which can be obtained by using the convolution integral with F(s) = H(t-3) and G(s) = sin(9(t-3)). They also clarified that in the convolution integral, the t's should be replaced with u,
  • #1
TyErd
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Homework Statement


I've attached the multiple choice question


Homework Equations





The Attempt at a Solution



I inversed laplaced the problem and i get an answer of 3 H(t-3) sin(9(t-3)) which isn't any of the options so I chose F but that was wrong. So far I've confidentally eliminated B,C and now F. A D and E remain. I tried using the convolution integral that yields me a different answer as well so I am really confused here
 

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  • #2
okay the answer is E. I already have 3 H(t-3) sin(9(t-3)). F(s) is H(t-3) and G(s) = sin(9(t-3)) but then when i write it as a convolution integral, shouldn't the t's be replaced with u instead of replacing the 3 as it is shown.
 
  • #3
it's cool i got it.
 

FAQ: What is the solution to the inverse Laplace transform of a given integral?

What is a Laplace transform integral?

A Laplace transform integral is a mathematical operation that converts a function of time into a function of complex frequency. It is often used in physics and engineering to analyze systems that involve differential equations.

How is a Laplace transform integral calculated?

A Laplace transform integral is calculated by taking the integral of a function multiplied by an exponential term, with a complex variable as the exponent. This integral can be solved using various techniques, such as partial fraction decomposition and the Laplace transform table.

What is the purpose of using a Laplace transform integral?

The purpose of using a Laplace transform integral is to simplify the analysis of complex systems that involve differential equations. It allows for the conversion of a time-domain function into a frequency-domain function, making it easier to solve and understand.

What are the advantages of using a Laplace transform integral?

One advantage of using a Laplace transform integral is that it allows for the solution of differential equations with initial conditions. It also helps in solving systems with multiple inputs and outputs, and it can be used to analyze the stability of a system.

What are some applications of Laplace transform integrals?

Laplace transform integrals have various applications in physics, engineering, and mathematics. They are used in circuit analysis, control systems, signal processing, and solving differential equations in mechanics and electromagnetics. They are also used in probability and statistics to calculate moments and generating functions.

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