Question

Given the Laplace transform F(s)= s ² +4s+5/3​ , what is the inverse Laplace transform f(t)?

A) f(t)=3e −²ᵗ cos(t)
B) f(t)=3e −²ᵗ sin(t)
C) f(t)=3e −²ᵗ cos(2t)
D) f(t)=3e ⁻ᵗ sin(t)
​

Answer 1

To find the inverse Laplace transform of F(s) = (s^2 + 4s + 5)/3, we use partial fraction decomposition. The inverse Laplace transform is f(t) = (1/3)(t^2 + 4t) + 5/3.

Step-by-step explanation:

To find the inverse Laplace transform of F(s) = (s^2 + 4s + 5)/3, we can use partial fraction decomposition. We can rewrite the expression as F(s) = (s^2 + 4s + 5)/3 = (s^2 + 4s)/3 + 5/3. This can be further simplified to F(s) = (1/3)(s^2 + 4s) + 5/3.

Using the linearity property of the Laplace transform, we can find the inverse Laplace transform of each term separately. The inverse Laplace transform of (1/3)(s^2 + 4s) is (1/3)(t^2 + 4t), and the inverse Laplace transform of 5/3 is 5/3. Therefore, the inverse Laplace transform of F(s) is f(t) = (1/3)(t^2 + 4t) + 5/3.