Question

"Compute the Laplace transforms of the following functions:

(a) f(t) = te³ᵗ
(b) f(t) = eᵗ sin(2t)"

Answer 1

The Laplace transform of f(t) = te³ᵗ is 1/(s²(s-3)). The Laplace transform of f(t) = eᵗ sin(2t) is 2/((s-1)(s²+4)).

Step-by-step explanation:

(a) To compute the Laplace transform of f(t) = te³ᵗ, we can use the property of linearity and the Laplace transform of the function e³ᵗ. The Laplace transform of e³ᵗ is 1/(s-3), where s is the complex variable. Using the property of linearity, we have:

L{te³ᵗ} = L{t} * L{e³ᵗ} = (1/s²) * (1/(s-3)) = 1/(s²(s-3)).

(b) To compute the Laplace transform of f(t) = eᵗ sin(2t), we can again use the property of linearity and the Laplace transforms of the functions eᵗ and sin(2t). The Laplace transform of eᵗ is 1/(s-1), and the Laplace transform of sin(2t) is 2/(s²+4). Using the property of linearity, we have:

L{eᵗ sin(2t)} = L{eᵗ} * L{sin(2t)} = (1/(s-1)) * (2/(s²+4)) = 2/((s-1)(s²+4)).