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Chapter 31: Problem 18

Find the inverse transforms of the given functions of \(s\). $$F(s)=\frac{6}{s^{2}+4}$$

Short Answer

Expert verified
The inverse Laplace transform of \( F(s) = \frac{6}{s^2 + 4} \) is \( f(t) = 3\sin(2t) \).

Step by step solution

01

Identify the Standard Laplace Transform

Recognize that the function of the form \( F(s) = \frac{a}{s^2 + a^2} \) is linked to a standard Laplace transform property. This format corresponds to the inverse Laplace transform of \( f(t) = a \sin(at) \).
02

Compare with the Standard Formula

Compare \( F(s) = \frac{6}{s^2 + 4} \) with the standard form \( \frac{a}{s^2 + a^2} \). Here, \( a^2 = 4 \), so \( a = 2 \). Therefore, \( F(s) = \frac{6}{s^2 + 2^2} \) matches the structure of the standard formula.
03

Apply the Inverse Laplace Transform

Use the identified standard inverse formula. Since \( a = 2 \) and the numerator is 6, which corresponds to \( 2\cdot 3 \), the inverse transform involves a constant multiplier. Thus \( \mathcal{L}^{-1}\left\{ \frac{6}{s^2 + 4} \right\} = 3\sin(2t) \).
04

Write the Final Inverse Transform Expression

Putting it all together, the inverse Laplace transform of the given function is \( f(t) = 3\sin(2t) \).

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Key Concepts

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

Laplace Transform
The Laplace Transform is a powerful tool used in engineering mathematics to simplify the process of solving differential equations. It transforms a complex differential equation in the time domain into a simpler algebraic equation in the s-domain.
  • This transformation makes it easier to manipulate and solve the equations.
  • It allows engineers and mathematicians to handle linear time-invariant systems efficiently.
Specifically, the Laplace Transform turns functions of time, like transients in electrical circuits or mechanical systems, into functions of complex frequency.

In essence, using the Laplace Transform reduces the complexity of solving differential equations by moving the operations into a different domain. Thus, it provides a simpler method to explore the behavior of dynamic systems.
Engineering Mathematics
Engineering mathematics relies heavily on advanced mathematical tools like the Laplace Transform to analyze and design systems.
  • These mathematical techniques help in modeling engineering problems that involve continuous-time signals or systems.
  • They play a crucial role in control systems, signal processing, fluid dynamics, and other fields.
Understanding and applying the Laplace Transform allows engineers to simplify complex problems into manageable calculations.

In practical applications, engineers use these transformations to design stable and efficient systems, whether in mechanical, electrical, or civil engineering. It bridges the gap between theoretical constructs and real-world applications.
Differential Equations
Differential equations form the backbone of modeling various physical systems, describing how they change with respect to time.
  • The solutions to differential equations can predict future behavior, optimize processes, and control systems.
  • Inverse Laplace Transforms offer a way to find these solutions systematically.
By converting differential equations into algebraic form using the Laplace Transform, one can easily apply inverse transformations to return to the time domain.

This method, as seen in the original exercise, involved transforming a function in the s-domain back to a time function, like transforming \( F(s) = \frac{6}{s^2 + 4} \) to \( f(t) = 3\sin(2t) \). This step is crucial in engineering to ensure real-time applicability and easy analysis of the system dynamics.

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Most popular questions from this chapter

Solve the given problems by solving the appropriate differential equation. The velocity \(v\) of a meteor approaching the earth is given by \(v \frac{d v}{d r}=-\frac{G M}{r^{2}},\) where \(r\) is the distance from the center of the earth, \(M\) is the mass of the earth, and \(G\) is a universal gravitational constant. If \(v=0\) for \(r=r_{0},\) solve for \(v\) as a function of \(r\)

Solve the given problems by solving the appropriate differential equation. The rate of change of the radial stress \(S\) on the walls of a pipe with respect to the distance \(r\) from the axis of the pipe is given by \(r \frac{d S}{d r}=2(a-S),\) where \(a\) is a constant. Solve for \(S\) as a function of \(r\)

Solve the given differential equations by Laplace transforms. The function is subject to the given conditions. A spring is stretched \(1 \mathrm{m}\) by a \(20-\mathrm{N}\) weight. The spring is stretched 0.5 m below the equilibrium position with the weight attached and then released. If it is in a medium that resists the motion with a force equal to \(12 v\), where \(v\) is the velocity, find the displacement y of the weight as a function of the time.

Find the inverse transforms of the given functions of \(s\). $$F(s)=\frac{3}{s^{4}+4 s^{2}}$$

Solve the given differential equations. $$y^{\prime \prime \prime}-y^{\prime}=\sin 2 x$$
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