Answer: The Laplace transform of the given function is 1 / (s - 2) + 1 / s².
Explanation:
The Laplace transform of a function is a mathematical tool used to convert a function in the time domain to a function in the complex frequency domain. To find the Laplace transform of the given function f(t)={e²ᵗ if 0≤t<1 and t if t≥1}, we need to split it into two parts based on the given condition.
For the first part, where 0≤t<1, the function is e²ᵗ. The Laplace transform of e²ᵗ is given by: L{e²ᵗ} = 1 / (s - 2)
For the second part, where t≥1, the function is t. The Laplace transform of t is given by: L{t} = 1 / s²
Therefore, the Laplace transform of the given function f(t)={e²ᵗ if 0≤t<1 and t if t≥1} is:
L{f(t)} = L{e²ᵗ} + L{t}
= 1 / (s - 2) + 1 / s²
So, the Laplace transform of the given function is 1 / (s - 2) + 1 / s².