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Chapter 6: Problem 6

Showing the details of your work, find \(\mathscr{L}(f)\) if \(f(t)\) equals: $$t^{2} \sin 3 t$$

Short Answer

Expert verified
\( \mathscr{L}(t^2 \sin 3t) = \frac{6}{(s^2 + 9)^3} \)

Step by step solution

01

Understand the Laplace Transform

The Laplace Transform of a function \( f(t) \), denoted as \( \mathscr{L}(f) \), is defined as \( \mathscr{L}(f) = \int_{0}^{\infty} e^{-st} f(t) \, dt \). It transforms a time-domain function into a complex frequency-domain function. Our goal is to apply this definition to \( f(t) = t^2 \sin 3t \).
02

Identify the Formula for Special Cases

For functions in the form \( t^n \sin(at) \), we use the formula for the Laplace Transform: \[ \mathscr{L}(t^n \sin(at)) = \frac{n! \cdot a}{(s^2 + a^2)^{n+1}} \]. Here, \( n = 2 \) and \( a = 3 \).
03

Substitute Known Values into the Formula

Substitute \( n = 2 \) and \( a = 3 \) into the formula: \[ \mathscr{L}(t^2 \sin 3t) = \frac{2! \cdot 3}{(s^2 + 3^2)^{3}} \].
04

Simplify the Expression

Calculate \( 2! = 2 \) and \( 3^2 = 9 \). Substitute these into the expression: \[ \frac{2 \cdot 3}{(s^2 + 9)^{3}} = \frac{6}{(s^2 + 9)^{3}} \].
05

Conclusion: Laplace Transform of the Given Function

The final result is \( \mathscr{L}(t^2 \sin 3t) = \frac{6}{(s^2 + 9)^{3}} \). This is the Laplace Transform of the function \( f(t) = t^2 \sin 3t \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

frequency-domain transformation
The concept of frequency-domain transformation is key to understanding the Laplace Transform. Essentially, the Laplace Transform is a technique used to change functions from the time domain to the frequency domain. This transformation is helpful because it simplifies the analysis and solution of differential equations.
  • When we talk about the frequency domain, we are primarily dealing with a complex plane with a real part and an imaginary part.
  • This transformation makes the function easier to analyze, especially for engineering and control systems applications.
  • The original time-domain function is often difficult to work with due to its complexity or oscillatory nature. Transforming it helps to highlight its frequency characteristics.
In this specific exercise, we employ the Laplace Transform to convert the given function, which is originally a time-domain function, into the frequency domain. This allows us to simplify the problem and derive a solution in a more straightforward manner.
time-domain function
A time-domain function is that which is expressed in terms of time, denoted usually by the variable \( t \). In this exercise, we are working with the time-domain function \( f(t) = t^2 \sin 3t \).
  • Time-domain functions are typically used to describe how a system evolves over time.
  • These functions might show trends such as growth, decay, or periodic oscillations like a sine wave.
  • The key challenge with time-domain functions is that their behavior can be complex, which makes them difficult to analyze directly when solving differential equations.
The piecewise nature of time domain functions often requires them to be converted into a more manageable format through transformations like the Laplace Transform. The function \( t^2 \sin 3t \) is an example of a time-domain function that involves both polynomial and trigonometric elements.
Mathematical functions
Mathematical functions are a broad category in mathematics, and they play a critical role in the application of the Laplace Transform. In this exercise, we specifically dealt with a function combining polynomial and trigonometric elements: \( t^2 \sin 3t \).
  • Polynomial functions, like \( t^2 \), involve variables raised to a power. They typically describe scenarios where growth or decay is present.
  • Trigonometric functions, such as \( \sin 3t \), describe periodic behaviour, often related to waves or oscillations.
  • When combined, these functions can model complex dynamic systems that appear in engineering and physics.
Understanding these types of functions individually is crucial to applying transformations. It ensures precision when dealing with complex equations and transformations, like those encountered in the Laplace Transform process. By comprehensively understanding each component of \( f(t) = t^2 \sin 3t \), one can more easily navigate through steps of finding the Laplace Transform.

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Most popular questions from this chapter

Solve the following initial value problems by the Laplace transform. (If necessary, use partial fraction expansion as in Example \(4 .\) Show all details.) $$y^{\prime}+\frac{1}{2} y=17 \sin 2 t, \quad y(0)=-1$$

Using the Laplece transform and showing the details of your work, solve the initial value problem: $$y_{1}^{\prime}=5 y_{1}+5 y_{2}-15 \cos t+27 \sin t$$$$y_{2}^{\prime}=-10 y_{1}-5 y_{2}-150 \sin t$$$$y_{1}(0)=2, \quad y_{2}(0)=2$$

Using differentiation, integration, s-shifting, or convolution (and showing the details), find \(f(t)\) if \(\mathscr{P}(f)\) equals: $$\frac{2}{(s-k)^{3}}$$

Given \(F(s)=\mathscr{L}(f)\)Show the details. \(\left(L, n, k, a,b\right.\) are constants.) $$\frac{1}{(s-\sqrt{3})(s+\sqrt{5})}$$

Find the Laplace transforms of the following functions. Show the details of your work. \((a, b, k, \omega, \theta\) are constants.) $$\cos 2 \pi t$$
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