How to Find the Inverse Laplace Transform of ( frac{(s-1)^3}{s^4} )

Introduction

The inverse Laplace transform is a crucial tool in solving differential equations and analyzing linear time-invariant systems. In this article, we will walk through a step-by-step method to find the inverse Laplace transform of a specific function: ( F(s) frac{(s-1)^3}{s^4} ).

Step-by-Step Solution

To find the inverse Laplace transform, we will follow a series of steps. Let's begin by simplifying the given expression and then finding the inverse Laplace transform of each component.

Step 1: Expand the Numerator

First, we expand the numerator ((s-1)^3). Using the binomial theorem, we get:

((s-1)^3 s^3 - 3s^2 3s - 1)

Thus, we can rewrite the function ( F(s) ) as:

[ F(s) frac{s^3 - 3s^2 3s - 1}{s^4} frac{s^3}{s^4} - 3frac{s^2}{s^4} 3frac{s}{s^4} - frac{1}{s^4} ]

This simplifies to:

[ F(s) frac{1}{s} - 3frac{1}{s^2} 3frac{1}{s^3} - frac{1}{s^4} ]

Step 2: Find the Inverse Laplace Transforms of Each Term

Next, we find the inverse Laplace transforms of each component term:

( mathcal{L}^{-1}left{ frac{1}{s} right} 1 )

( mathcal{L}^{-1}left{ -frac{3}{s^2} right} -3t )

( mathcal{L}^{-1}left{ frac{3}{s^3} right} frac{3}{2}t^2 )

( mathcal{L}^{-1}left{ -frac{1}{s^4} right} -frac{1}{6}t^3 )

Combining these results, we get:

[ mathcal{L}^{-1}{ F(s) } 1 - 3t frac{3}{2}t^2 - frac{1}{6}t^3 ]

Final Result

Therefore, the inverse Laplace transform of ( F(s) frac{(s-1)^3}{s^4} ) is:

[ f(t) 1 - 3t frac{3}{2}t^2 - frac{1}{6}t^3 ]

Related Content

For a more in-depth understanding of inverse Laplace transforms and their applications, consider exploring the following related topics:

Inverse Laplace Transform

Laplace Transform

Partial Fraction Expansion

By delving into these subjects, you will gain a comprehensive understanding of the Laplace transform and its inverse, which are essential tools for solving differential equations and analyzing various physical systems.