Final answer:
To find the zero input response of a second-order LTIC system with the given initial conditions, solve the characteristic equation to determine the system's response, then apply the initial conditions to find the coefficients in the solution.
Step-by-step explanation:
The question involves an Linear Time-Invariant Continuous (LTIC) system specified by the second-order differential equation (D^2 + 10D + 25) y(t) = (D + 8) f(t), and asks to find the zero input response given initial conditions y(0) = 2, and y′(0) = 0. To determine the zero input response, we ignore the input function f(t), as the zero input response solely depends on the system's response to its initial conditions, not any external input.
Start by solving the characteristic equation associated with the differential equation, which is the same as the part multiplying the output y(t), hence m^2 + 10m + 25. Factoring this yields (m + 5)^2, indicating a repeated root at m = -5. Thus, the homogeneous solution to the differential equation is of the form y(t) = (C1 + C2 t) e^{-5t} where C1 and C2 are constants determined by the initial conditions.
To find these constants, apply the initial conditions: y(0) = 2 gives C1 = 2. Differentiating the equation to find y′(t) and then using y′(0) = 0, we find that C2 = 0. Consequently, the zero input response is y(t) = 2e^{-5t}.