Inverse Laplace Transform of Rational Functions: A Detailed Guide

In this article, we will thoroughly explore the process of finding the inverse Laplace transform of rational functions, specifically focusing on the function frac{s^2 6s 14}{s^3 s^2 4s - 4}. We will break down the steps involved in solving this problem, including factoring the denominator, using partial fraction decomposition, and determining the inverse Laplace transform of each simpler fraction.

Step 1: Factor the Denominator

The first step in solving the problem is to factor the denominator of the given rational function. The denominator is s^3 s^2 4s - 4. To factor this cubic polynomial, we can use the Rational Root Theorem to test potential rational roots. After testing several candidates, we determine that s 1 is a root. This leads to the factorization:

s^3 s^2 4s - 4 (s - 1)(s^2 2s 4)

Step 2: Partial Fraction Decomposition

Next, we express the original function using partial fractions:

frac{s^2 6s 14}{(s 1)(s^2 2s 4)} frac{A}{s 1} frac{Bs C}{s^2 2s 4}

Step 3: Solve for Coefficients

Multiplying through by the denominator, we get:

s^2 6s 14 A(s^2 2s 4) (Bs C)(s 1)

Expanding and collecting like terms will lead to a system of equations to solve for A, B, and C. The process involves equating coefficients and solving the resulting system:

A B 1 2A - B C 6 4A - C 14

By solving these equations, we determine:

A 3 B -2 C -2

Step 4: Rewrite the Fraction

With the coefficients determined, we can rewrite the original fraction as:

frac{3}{s 1} frac{-2s 2}{s^2 2s 4}

Step 5: Inverse Laplace Transform

Finally, we find the inverse Laplace transforms for each simpler fraction:

frac{3}{s 1} transforms to 3e^t frac{-2s 1}{s 1^2 3} can be split into two parts: -2e^{-t}cos(sqrt{3}t) and -frac{2}{sqrt{3}}e^{-t}sin(sqrt{3}t)

Combining everything, the final result for the inverse Laplace transform is:

3e^t - 2e^{-t}cos(sqrt{3}t) - frac{2}{sqrt{3}}e^{-t}sin(sqrt{3}t)

Understanding the inverse Laplace transform is crucial in many engineering and scientific applications. This guide demonstrates the systematic approach to solving such problems, ensuring clarity and accuracy in the calculations.