Final answer:
To find the system response y(t) for x(t) = u(t-1) based on the given transfer function H(s), we need to take the inverse Laplace transform of H(s) and multiply it by the Laplace transform of x(t). The system response y(t) is (4e^(-t) - 6e^(-2t)) * u(t-1). The impulse response is (4e^(-t) - 6e^(-2t)) * δ(t-1).
Step-by-step explanation:
To find the system response y(t) for x(t) = u(t-1) based on the given transfer function H(s), we need to take the inverse Laplace transform of H(s) and multiply it by the Laplace transform of x(t). The inverse Laplace transform of H(s) = 10/(s+1)(s+2) can be found using partial fraction decomposition. Let's assume A and B as the constants:
H(s) = A/(s+1) + B/(s+2)
To find A and B, we can multiply both sides by the denominator (s+1)(s+2) and replace s with -1 and -2, respectively.
By solving these equations, we find that A = 4 and B = -6.
So, the inverse Laplace transform of H(s) is h(t) = 4e^(-t) - 6e^(-2t).
Next, we need to find the Laplace transform of x(t) = u(t-1). The Laplace transform of u(t-a) is given by U(s) = e^(-as)/s. By substituting a = 1, we get U(s) = e^(-s)/s.
Finally, to find y(t), we convolve h(t) and U(s). The convolution of two functions f(t) and g(t) in the time domain is equivalent to multiplying their Laplace transforms F(s) and G(s) in the Laplace domain.
In this case, the Laplace transform of y(t), Y(s), is the product of H(s) and U(s).
Y(s) = H(s) * U(s) = (4e^(-s) - 6e^(-2s)) * (e^(-s)/s)
Now we need to take the inverse Laplace transform of Y(s) to obtain y(t).
Using the convolution theorem, we can rewrite Y(s) as the product of the Laplace transform of h(t) and the Laplace transform of x(t).
Y(s) = H(s) * U(s) = H(s) * X(s)
By taking the inverse Laplace transform of Y(s), we find that y(t) = [inverse Laplace transform of H(s)] * x(t) = h(t) * x(t).
Therefore, the system response y(t) for x(t) = u(t-1) is y(t) = (4e^(-t) - 6e^(-2t)) * u(t-1).
To find the impulse response, we substitute x(t) with the impulse function δ(t) = 1 in the equation.
The impulse response h(t) is y(t) = (4e^(-t) - 6e^(-2t)) * δ(t-1).