Question

Find the inverse Laplace transform of (s+4)/(s(s-1)(s²+4))

Answer 1

The inverse Laplace transform of (s+4)/(s(s-1)(s²+4)) is (
`e^t`+ 2cos(2t) - sin(t) - cos(t))/5.

Step-by-step explanation:

To find the inverse Laplace transform of (s+4)/(s(s-1)(s²+4)), we can use partial fraction decomposition. First, express the given expression as A/s + B/(s-1) + (Cs + D)/(s²+4), where A, B, C, and D are constants. After solving for these constants, we obtain the partial fraction decomposition as (1/5)/s + (1/5)/(s-1) + (
`e^t/5` + 2cos(2t)/5 - sin(t)/5 - cos(t)/5)/(s²+4).

Now, we can easily find the inverse Laplace transform of each term. The inverse Laplace transform of 1/s is 1, 1/(s-1) is
`e^t`, and 1/(s²+4) is (1/2)sin(2t). For the last term involving the exponential and trigonometric functions, we use the properties of the Laplace transform to arrive at the inverse Laplace transform (
`e^t` + 2cos(2t) - sin(t) - cos(t))/5.

In summary, the final answer is (
`e^t` + 2cos(2t) - sin(t) - cos(t))/5, obtained through partial fraction decomposition and the inverse Laplace transforms of individual terms.