Final answer:
The impulse response of a system defined by the differential equation (D² + 4D + 3)y(t) = (D+5)x(t) is determined by solving the corresponding homogeneous equation, finding its characteristic equation, and then applying boundary conditions related to unit impulse behavior at t=0.
Step-by-step explanation:
The student has asked to find the unit impulse response of a system specified by the differential equation (D² + 4D + 3)y(t) = (D+5)x(t), where D represents the derivative with respect to time.
To find the unit impulse response, also known as the Green's function or impulse response function, we need to solve the equation when x(t) is a delta function, δ(t). The solution to the homogeneous equation (D² + 4D + 3)y(t) = 0 provides the natural modes of the system, and the particular solution will give us the impulse response.
First, we solve the characteristic equation associated with the homogeneous part: λ² + 4λ + 3 = 0. The solutions to this quadratic equation, λ_1 and λ_2, will help us express the solution in the form y(t) = C_1e^{λ_1t} + C_2e^{λ_2t}.
After determining the constants based on boundary conditions, mainly that the response to a unit impulse is initially zero but its derivative is infinite at t=0 (due to the impulse), we can determine the full impulse response of the system, ensuring it satisfies the given differential equation with x(t) = δ(t).