Question

Inverse Laplace using partial fractions:

F(s) = 8/(s^2(s^2 - s - 2))
F(s) = 2s/((s - 2)(s + 1)(s + 3))

Answer 1

The decomposition of the given Laplace transform functions into partial fractions facilitates their inverse transformation back into the time domain, which serves to solve linear differential equations.

Step-by-step explanation:

Inverse Laplace Transformation Using Partial Fractions

To find the inverse Laplace transform using partial fractions, we must first decompose the given function into simpler fractions that can be easily transformed back into the time domain. For the function F(s) = 8/(s^2(s^2 - s - 2)), we would need to factor the denominator and then express it as a sum of simpler fractions. Each term corresponds to a specific form in the Laplace table, allowing for an inverse transformation.

The second function F(s) = 2s/((s - 2)(s + 1)(s + 3)) can be similarly decomposed into partial fractions. After finding the constants for each fraction, we apply the inverse Laplace transform to find the solution in the time domain.

Using graphing calculators or algebraic manipulation to find the constants in partial fractions is a common method applied to linear differential equations and can simplify solving for unknown values. Once the fractions are obtained, the time domain solutions typically involve exponential functions and possibly sinusoidal components, which represent the system's response over time.