Question

Find f(t) for each of the following fuctions:

F(s) = 8s² + 37s + 32 / (s +1)(s+2)(s+4)

Answer 1

Finding f(t) for the given function F(s) involves partial fraction decomposition and inverse Laplace transformation. Here's a summary of the steps and results:

`f(t) = 8e^((-t)) + 19e^((-2t)) - 5e^((-4t))`

Step-by-step explanation:

Partial Fraction Decomposition:

We first decompose F(s) into partial fractions with unknown coefficients A, B, and C:

F(s) = A/(s+1) + B/(s+2) + C/(s+4)

Then, we solve for A, B, and C by multiplying both sides by the common denominator and equating coefficients of like terms. This yields:

A = 8, B = 19, C = -5

Inverse Laplace Transformation:

Next, we apply the inverse Laplace transform to each term in the decomposed fraction:

`f(t) = A * L^{-1({1/(s+1)})} + B * L^{-1{(1/(s+2)})} + C * L^{-1({1/(s+4)})}`

Using the table of Laplace transforms, we obtain:

`f(t) = 8e^((-t)) + 19e^((-2t)) - 5e^((-4t))`

Therefore, f(t) is expressed as a combination of three exponential terms with coefficients determined by the partial fraction decomposition.