Final answer:
To find the partial Laplace transform of the function f(t) = 2 for 0 ≤ t ≤ 2 and 0 otherwise, integrate 2e^{-st} from 0 to 2, resulting in -2/s [e^{-2s} - 1]. This represents the partial Laplace transform over the interval where f(t) is non-zero.
Step-by-step explanation:
The question is asking to compute the partial Laplace transform of a piecewise function f(t) that has a value of 2 in the interval [0,2] and 0 otherwise.
The partial Laplace transform of f(t) is found by integrating the product of f(t) and e-st from t=0 to t=2, since the function is zero for t > 2.
The integral we need to evaluate is:
∫02 2e-st dt
By solving this integral, we obtain:
2s[e-st] 02 = 2s [e-2s - 1]
This result represents the partial Laplace transform of the function f(t) over the interval where it is non-zero.