Understanding the Laplace Inverse of ( frac{s}{s^2 a} ) Using Trigonometric Functions

In the realm of engineering and mathematics, the Laplace Transform is a fundamental tool for analyzing and solving linear differential equations. Understanding the inverse Laplace Transform is crucial for practical applications. In this article, we explore the solution to the problem ( mathcal{L}^{-1} left[frac{s}{s^2 a} right] ), where ( a ) is a constant. We'll use trigonometric functions to derive the solution, and we'll reference the provided mathematical expressions for clarity.

Introduction to Laplace Transform and Inverse Laplace Transform

The Laplace Transform is a mathematical technique that converts a function of time, ( f(t) ), into a function of the complex variable ( s ). The inverse Laplace Transform, denoted by ( mathcal{L}^{-1} ), is the reverse process, converting a function of ( s ) back into the time domain.

The Given Problem: ( mathcal{L}^{-1} left[frac{s}{s^2 a} right] )

The specific problem we are dealing with is the inverse Laplace Transform of the function ( frac{s}{s^2 a} ). This function can be simplified and transformed using properties and formulas of Laplace Transforms and inverse Laplace Transforms.

Simplifying the Given Function

First, let's rewrite the function in a more manageable form. Given:

[ Fs frac{s}{s^2 a} frac{s}{a s^2} frac{1}{a s} ]

However, we will use the given reference to refine the problem step by step:

[ Fs frac{s1}{s^2 s1} frac{s frac{1}{2}}{left(frac{1}{2}right)^2 left(frac{3}{4}right)} frac{frac{1}{2}}{left(frac{1}{2}right)^2 left(frac{3}{4}right)} mathcal{L}^{-1} left[ frac{frac{1}{2}}{left(frac{1}{2}right)^2 left(frac{3}{4}right)} right] frac{1}{2} cdot frac{2}{sqrt{3}} mathcal{L}^{-1} left[ frac{frac{sqrt{3}}{2}}{left(frac{1}{2}right)^2 left(frac{sqrt{3}}{2}right)^2} right] ]

Step-by-Step Solution

Let's break down the steps to solve the inverse Laplace Transform:

Step 1: Simplify the Given Function

Using the provided reference, we can simplify the function in terms of known Laplace Transform properties and results.

[ Fs frac{s1}{s^2 s1} frac{frac{1}{2} s}{left(frac{1}{2}right)^2 left(frac{3}{4}right)} ]

Step 2: Apply Known Inverse Laplace Transform Properties

From Table of Laplace Transforms and known properties, we can use the following:

The Inverse Laplace Transform of ( frac{s}{s^2 a^2} ) is ( cos(at) ). The Inverse Laplace Transform of ( frac{a}{s^2 a^2} ) is ( sin(at) ).

Given these properties, we can transform the given function into a form that matches the known inverse Laplace Transforms.

Step 3: Final Form

Finally, using the properties and simplifications, we can write the solution as:

[ mathcal{L}^{-1} left[ frac{frac{1}{2} s}{left(frac{1}{2}right)^2 left(frac{3}{4}right)} right] e^{-frac{1}{2}t} cosleft(frac{sqrt{3}}{2} tright) frac{1}{sqrt{3}} e^{-frac{1}{2}t} sinleft(frac{sqrt{3}}{2} tright) ]

Conclusion

In this article, we explored the inverse Laplace Transform of a function involving trigonometric functions. The process involved simplifying the given function and using known properties and results from the Table of Laplace Transforms. Understanding these techniques is essential for solving complex differential equations and analyzing systems in engineering and physics.

References and Further Reading

For a deeper understanding, you can refer to the following links:

Wolfram Alpha Symbolab Tables of Laplace Transforms (available online)