Question

Determine $L^{-1}[F]$. $$F(s)=\frac{4}{(s+2)^{3}}$$.

   Determine $L^{-1}[F]$.
$$F(s)=\frac{4}{(s+2)^{3}}$$.

Differential Equations and Linear Algebra

Differential Equations and Linear Algebra

Stephen W. Goode,… 4th Edition

Chapter 10, Problem 29

Instant Answer

Solved by Expert Sriram Soundarrajan

06/25/2020

Step 1

The general form of a Laplace transform is $F(s) = \frac{n!}{(s-a)^{n+1}}$, which corresponds to $f(t) = e^{at}t^n$ in the time domain. Show more…

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Determine $L^{-1}[F]$. $$F(s)=\frac{4}{(s+2)^{3}}$$.

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Key Concepts

Exponential Shift Theorem

The exponential shift theorem, also known as the first shifting theorem, states that a shift in the s variable corresponds to multiplication by an exponential function in the time domain. This theorem is fundamental when dealing with shifted arguments in the Laplace domain and allows for the straightforward inversion of transforms that include terms of the form (s+a) raised to a power.

Standard Laplace Transform Formulas for Power Functions

There are standard formulas for the Laplace transforms of power functions multiplied by an exponential, such as t^n e^{-at}. For example, the formula L{t^n e^{-at}} = n!/(s + a)^(n+1) provides a direct way to match a given Laplace transform to the time-domain function, thereby facilitating the computation of the inverse Laplace transform when the transformed function is presented in a format that fits this pattern.

Laplace Transform

The Laplace transform is an integral transform that converts a function of time, typically representing a signal or a system’s response, into a function of a complex variable. This transformation is useful for solving differential equations and analyzing linear time-invariant systems because it turns differentiation and integration in the time domain into algebraic operations in the s-domain.

Inverse Laplace Transform

The inverse Laplace transform reverses the process of the Laplace transform, recovering the original time-domain function from its s-domain representation. It is essential for interpreting the results of solutions obtained in the s-domain and converting them back into a form that can be used for analyzing real-world behaviors.

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