(9) (a) If f(t) = x1e −kt cos(ωt) +x2e −kt sin(ωt), then f'(t) has the same formwith coefficients y1, y2.
(i) Find the matrix A such that y = Ax.
(ii) Find the characteristic polynomial of A.
(iii) What can you say about eigenvalues of A? (iv) Interpretyour answer to
(iii) as a calculus statement. That is, explain how your answerto (iii) could have been predicted from a basic fact ofcalculus.
(b) If f(t) = x1 sin(t) + x2 cos(t) +x3tsin(t) + x4t cos(t), then f'(t) has thesame form with coefficients y1, ..., y4. Same questions (a)-(d) aspreviously.
