Question

Find the inverse laplace transform of () = 3 2 6 9

Answer 1

The inverse Laplace transform of
`\(F(s) = (3s^2 + 6)/(9)\) is \(f(t) = t^2 + (2)/(3)\)` due to the linearity property of the Laplace transform.

Step-by-step explanation:

The given function
`\(F(s) = (3s^2 + 6)/(9)\)` can be rewritten as
`\(F(s) = (1)/(3)\left(s^2 + 2\right)\)`. Using the linearity property of the Laplace transform, we can split it into two terms:
`\((1)/(3)L\{s^2\} + (1)/(3)L\{2\}\)`. The inverse Laplace transform of
`\((1)/(3)L\{s^2\}\) is \(t^2\)`, and the inverse Laplace transform of
`\((1)/(3)L\{2\}\) is \((2)/(3)\)` due to the basic Laplace transform properties.

Therefore, the overall inverse Laplace transform is
`\(f(t) = t^2 + (2)/(3)\)`. This result is obtained by considering each term individually and applying the corresponding inverse Laplace transform. The linearity property allows us to decompose the expression into simpler terms and find their inverse transforms independently.

In conclusion, the inverse Laplace transform of the given function is
`\(f(t) = t^2 + (2)/(3)\)`, and this result is derived by leveraging the linearity property of the Laplace transform, which facilitates the analysis of each term separately and obtaining the inverse transform of the entire expression.