Understanding the Inverse Laplace Transform of ( mathcal{L}^{-1} left( frac{s}{s^2 2^2} right) )

Introduction to the Inverse Laplace Transform

The Inverse Laplace Transform, denoted by ( mathcal{L}^{-1} ), is a crucial tool in solving linear differential equations and analyzing systems in control theory and signal processing. It transforms a function of the Laplace domain back to the time domain. The Laplace Transform, ( mathcal{L} ), converts a time-domain function to a complex frequency domain function, which can simplify the process of solving differential equations.

The Problem at Hand

We are tasked with finding the Inverse Laplace Transform of the function ( frac{s}{s^2 2^2} ). This function represents a transfer function or a transform in the Laplace domain, and its inverse is of great interest.

Step-by-Step Solution

Let's break down the problem step-by-step:

Step 1: Decompose the Function

First, we decompose the function ( frac{s}{s^2 2^2} ) into simpler components. Notice that ( s^2 2^2 ) can be written as ( s^2 4 ).

[ mathcal{L}^{-1} left( frac{s}{s^2 4} right) mathcal{L}^{-1} left( frac{s}{s^2 2^2} right) ]

Step 2: Partial Fraction Decomposition

Next, we can use the formula for the Inverse Laplace Transform of a term of the form ( frac{s}{s^2 a^2} ). The Inverse Laplace Transform of ( frac{s}{s^2 a^2} ) is ( cos(at) ) and the Inverse Laplace Transform of ( frac{a}{s^2 a^2} ) is ( sin(at) ).

Step 3: Express as a Difference of Two Terms

Given the form ( frac{s}{s^2 2^2} ), we can write:

[ frac{s}{s^2 4} frac{s}{s^2 2^2} frac{1}{2} left( frac{2s}{s^2 4} right) frac{1}{2} left( frac{s^2 4 - 4 2s}{s^2 4} right) frac{1}{2} left( 1 - frac{4}{s^2 4} right) ]

Thus:

[ mathcal{L}^{-1} left( frac{s}{s^2 2^2} right) frac{1}{2} left[ mathcal{L}^{-1} left( frac{1}{s^2 4} right) - 2 mathcal{L}^{-1} left( frac{2}{s^2 4} right) right] ]

Step 4: Apply the Inverse Laplace Transform

Now, applying the Inverse Laplace Transform to each term:

[ mathcal{L}^{-1} left( frac{1}{s^2 4} right) frac{1}{2} sin(2t) ]

[ mathcal{L}^{-1} left( frac{2}{s^2 4} right) sin(2t) ]

Hence:

[ mathcal{L}^{-1} left( frac{s}{s^2 4} right) frac{1}{2} left[ frac{1}{2} sin(2t) - 2 sin(2t) right] frac{1}{2} left[ -frac{3}{2} sin(2t) right] -frac{3}{4} sin(2t) ]

Conclusion

In conclusion, the Inverse Laplace Transform of ( frac{s}{s^2 2^2} ) is ( -frac{3}{4} sin(2t) ). This result is obtained by breaking down the original function, applying known Inverse Laplace Transform properties, and simplifying the expression.

Additional Insight

Understanding the Inverse Laplace Transform can help in solving more complex differential equations and analyzing various engineering and scientific systems. The step-by-step approach we used can be applied to similar problems in the future, making the process less daunting.

Keywords

Inverse Laplace Transform, Laplace Transform, Differential Equations