Final answer:
To find the system response using the Laplace transform, apply it to the given differential equation and solve for Y(s). The transfer function H(s) can then be derived from the ratio of the output to the input in the Laplace domain.
Step-by-step explanation:
Response to the Differential Equation Using Laplace Transform
To find the response y(t) of the system described by the differential equation dy(t)/dt + 2y(t) = dx(t)/dt with x(t) = e⁻¹u(t) and initial condition y(0-) = 21/2, we use the Laplace transform. First, assume u(t) is the unit step function. The Laplace transform of x(t) is thus X(s) = L{e⁻¹u(t)} = 1/(s+1). Applying the Laplace transform to the differential equation yields L{dy(t)/dt} + 2L{y(t)} = L{dx(t)/dt}. Using the property L{dy(t)/dt} = sY(s) - y(0-), and substituting the initial condition and X(s), we have sY(s) - 21/2 + 2Y(s) = sX(s) - x(0-). Solve this equation for Y(s) to obtain the Laplace transform of the output y(t).
Finding the Transfer Function H(s)
The transfer function H(s) of the system is found by taking the ratio of the output Y(s) to the input X(s) when the initial conditions are zero. Since the system is linear and time-invariant, H(s) = Y(s) / X(s) = (s / (s+2)) once we clear the initial conditions. This transfer function represents the system's response in the frequency domain.