Final answer:
The Laplace transform of the function v(t) = (20e⁻⁴ᵗ + 25e⁻⁵ᵗ) u(t) is L{v(t)} = 20 / (s + 4) + 25 / (s + 5).
Step-by-step explanation:
The question asks for the Laplace transform of a given function, v(t) = (20e⁻⁴ᵗ + 25e⁻⁵ᵗ) u(t), where t > 0. The u(t) denotes the Heaviside step function, which is 1 for t > 0, indicating the function v(t) is defined for t > 0. To find the Laplace transform, we use the basic transform pair L{e⁻ᵗᵗ} = 1 / (s + a), and apply it to each term of v(t).
The transforms for the individual terms are:
- L{20e⁻⁴ᵗ} = 20 / (s + 4)
- L{25e⁻⁵ᵗ} = 25 / (s + 5)
Adding these together gives us the Laplace transform of v(t):
L{v(t)} = 20 / (s + 4) + 25 / (s + 5)