Understanding the Inverse Laplace Transform of s - 5 / s^2 6s 13

The Inverse Laplace Transform is a fundamental concept in the field of engineering and applied mathematics, particularly when solving differential equations. One important application involves finding the inverse Laplace transform of complex expressions. In this article, we will explore how to determine the inverse Laplace transform of the expression s - 5 / s^2 6s 13.

Tackling the Denominator

The first step is to manipulate the denominator of the given expression s - 5 / s^2 6s 13. We start by completing the square for the denominator:

s^2 6s 13 can be rewritten as s^2 6s 9 4, which simplifies to (s 3)^2 4. This manipulation helps us fit the denominator into a standard form that can be easily transformed.

Thus, the expression becomes:

s - 5 / (s 3)^2 4

Decomposition and Transformation

The next step is to decompose the expression into more manageable parts. Notice that:

s - 5 (s 3) - 8

So:

(s - 5) / (s^2 6s 13) (s 3 - 8) / (s 3)^2 4

This further breaks down into:

(s 3) / (s 3)^2 4 - 8 / (s 3)^2 4

Applying the Laplace transform properties, we can find the inverse Laplace transform of each part separately:

Standard Transformations

For the term (s 3) / (s 3)^2 4, we use the property:

mathscr{L}^{-1} [ (s a) / (s a)^2 b^2 ] e^{-at} cos(bt)

Here, a 3 and b 2. Therefore:

mathscr{L}^{-1} [ (s 3) / (s 3)^2 4 ] e^{-3t} cos(2t)

For the term -8 / (s 3)^2 4, we use the property:

mathscr{L}^{-1} [ b / (s a)^2 b^2 ] b e^{-at} sin(bt)

Again, with a 3 and b 2:

mathscr{L}^{-1} [ -8 / (s 3)^2 4 ] -4 e^{-3t} sin(2t)

Final Inverse Laplace Transform

Combining both parts, we get:

mathscr{L}^{-1} [(s - 5) / (s^2 6s 13)] e^{-3t} cos(2t) - 4 e^{-3t} sin(2t)

Conclusion

The process outlined above demonstrates a systematic approach to finding the inverse Laplace transform of complex expressions. By breaking down the expression into simpler components and applying well-known Laplace transform properties, we can easily determine the desired result. This method is particularly useful in solving differential equations and analyzing various physical systems in engineering and physics.

Related Keywords and Phrases

Inverse Laplace Transform Laplace Transform Mathematical Functions