Inverse Laplace Transform via Completing the Square

Find the inverse Laplace transform of $\frac{s}{s^2 - 2s + 5}$ using the method of completing the square.

The task of finding the inverse Laplace transform of a given function often revolves around recognizing the standard forms that directly relate to known transforms and utilizing strategic algebraic manipulations to fit these forms. In this problem, the primary technique employed is completing the square, a method particularly useful in simplifying quadratic expressions in the Laplace transform's denominator. By rewriting the quadratic in the form of a squared binomial plus or minus a constant, you can more clearly identify the appropriate inverse transform based on standard Laplace transform pairs.

Completing the square is crucial because it allows the transformation of a complex expression into a more manageable form, potentially revealing connections to well-documented inverse transforms such as those involving exponential decays or sinusoidal functions. This also aids in delineating transient and steady-state behaviors in the solutions to differential equations, especially when translating between time and frequency domains.

Furthermore, understanding and applying the method of completing the square in the context of Laplace transforms also provides deeper insights into the stability and dynamic response of systems modelled by differential equations. Here, seeing the problem through the lens of frequency domain solutions allows for a comprehensive understanding of how various components behave over time, invaluable for fields such as engineering and physics.