Final answer:
The unit impulse responses of the given system equations are: a) u(t), b) e^(-it) + e^(it), c) e^(-2t), d) e^(-3t).
Step-by-step explanation:
To find the unit impulse response of a system, we need to find the inverse Laplace transform of the transfer function. Let's go through each option:
a) H(s) = 1/s
The inverse Laplace transform of 1/s is the unit step function, u(t).
b) H(s) = s/(s² + 1)
Using partial fraction decomposition, we can write the transfer function as H(s) = 1/(s + i) + 1/(s - i), where i is the imaginary unit. The inverse Laplace transform of 1/(s + i) is e^(-it), and the inverse Laplace transform of 1/(s - i) is e^(it). Therefore, the unit impulse response is e^(-it) + e^(it).
c) H(s) = e^(-2s)
The inverse Laplace transform of e^(-2s) is the unit impulse function scaled by a factor of e^(-2t), where t is the time variable. Therefore, the unit impulse response is e^(-2t).
d) H(s) = 1/(s + 3)
The inverse Laplace transform of 1/(s + 3) is e^(-3t). Therefore, the unit impulse response is e^(-3t).