Question

A Consider an LTIC system specified by the differential equation

(D²+4D+k)y(t)=(3D+5)x(t)

Using initial conditions y₀(t)=3 and y₀(t)=−7, apply MATLAB's dsolve command to determine the zero-input response when:
(a) k=3,
(b) k=4,
(c) k=40.

Answer 1

To determine the zero-input response of an LTIC system for different values of k using MATLAB's dsolve command, the differential equation is solved with the given initial conditions, assuming zero input. Repeating the command for k=3, k=4, and k=40 .

Step-by-step explanation:

The student is asking about determining the zero-input response of an LTIC system described by a second-order differential equation. Using MATLAB's dsolve command, the zero-input response can be found by solving the homogeneous part of the differential equation given initial conditions for specific values of k. The given initial conditions are y0(t)=3 (initial value) and y0'(t)=-7 (initial derivative).

For each value of k, the differential equation will be solved using MATLAB assuming zero input, which means that the right side of the equation, which depends on x(t), will be set to zero. In MATLAB, the command will resemble:

syms y(t) D
D = diff;
y0 = 3;
y1 = -7;
// For k = 3
k = 3;
DEq = D^2*y + 4*D*y + k*y == 0;
ySol = dsolve(DEq, y(0) == y0, Dy(0) == y1);
// Repeat the procedure for k = 4 and k = 40

This MATLAB code will provide a symbolic solution for the zero-input response of the LTIC system for each specified value of k. By considering different values of k, the response of the system can be analyzed in underdamped, critically damped, or overdamped scenarios.