Question

Determine the inverse La Place transform of:

F₁(S) = (6s² + 8s + 3) / (s(s² + 2s + 5))

Answer 1

The inverse Laplace transform of F₁(S) = (6s² + 8s + 3) / (s(s² + 2s + 5)) can be determined using partial fraction decomposition and the Laplace transform table. The final inverse Laplace transform is f₁(t) = A + (B/2) e^(-t) cos(2t) + (C/2) e^(-t) sin(2t).

Step-by-step explanation:

The inverse Laplace transform of F₁(S) = (6s² + 8s + 3) / (s (s² + 2s + 5)) can be found using partial fraction decomposition and standard Laplace transform table.

Step 1: Perform partial fraction decomposition on F₁(S)

F₁(S) = A/s + (Bs + C)/ (s² + 2s + 5)

Step 2: Solve for A, B, and C using algebraic manipulation.

Step 3: Use the Laplace transform table to find the inverse transform of each term.

Step 4: Combine the inverse transforms to obtain the final inverse Laplace transform.

Therefore, the inverse Laplace transform of F₁(S) is f₁(t) = A + (B/2) e^(-t) cos(2t) + (C/2) e^(-t) sin(2t)