Consider the following linear, time-invariant system:
The vector x = [x1 x2] T is known as the state vector. The parameter k0 is a constant. The input to the system is u. Develop a script using MathScript to perform the following computations:
(a) For 0 ≤ k0 ≤ 5, compute the eigenvalues of A. The eigenvalues can be complex (imaginary) numbers. Generate a plot of the real part of the eigenvalues versus the imaginary part.
(b) With the input u = 2 for t ≥ 0 and k = 0.5, find the solution using numerical integration and plot x versus t for 0 ≤ t ≤ 20. Use the initial conditions x(0) = [1 0] T