Final answer:
To find the inverse Laplace transform of 1 / (s²(s² + 4)), we can use partial fraction decomposition and a table of Laplace transforms.
Step-by-step explanation:
To find the inverse Laplace transform of the given function, 1 / (s²(s² + 4)), we can use partial fraction decomposition and a table of Laplace transforms.
The function can be rewritten as 1 / (s²(s² + 2²)). From the partial fraction decomposition, we get the following expression:
1 / (s²(s² + 2²)) = A / s + B / s² + C / (s - 2i) + D / (s + 2i)
Solving for the unknowns A, B, C, and D, we find that A = 1/4, B = -1/4, C = -1/8i, and D = 1/8i.
Now, we can use the table of Laplace transforms to find the inverse transform of each term. The inverse Laplace transform of A / s is simply A, the inverse Laplace transform of B / s² is Bt, and the inverse Laplace transform of C / (s - 2i) and D / (s + 2i) are Ce^(2it) and De^(-2it), respectively.