Final answer:
The system response y(t) is the inverse Laplace transform of the transfer function H(s). For H(s) = 2/(s² + 3s + 2), the response is y(t) = 2e-t - 2e-2t, which is the impulse response of the system.
Step-by-step explanation:
To determine the system response y(t) for an LTI system with the transfer function H(s) = 2/(s² + 3s + 2), we start by finding the inverse Laplace transform of H(s). This will give us the impulse response h(t) of the system. The poles of H(s) are the roots of the denominator s² + 3s + 2, which factorizes to (s + 1)(s + 2).
Therefore, the partial fraction decomposition of H(s) is:
H(s) = 2/((s+1)(s+2)) = A/(s+1) + B/(s+2)
By finding the values of A and B, we can determine the inverse Laplace of each term individually. After finding A and B, we get:
H(s) = 2/(s+1) - 2/(s+2)
The inverse Laplace transform gives us the impulse response:
h(t) = 2e-t - 2e-2t
This impulse response h(t) represents the system response y(t) to a delta function input. If the input were something other than a delta function, you would need to convolve this impulse response with the actual input to get the output y(t).