Chapter 3.4, Problem 5E

In Exercises 3-8, each linear system has complex eigenvalues. For each system,
(a) find the eigenvalues;
(b) determine if the origin is a spiral sink, a spiral source, or a center;
(c) determine the natural period and natural frequency of the oscillations,
(d) determine the direction of the oscillations in the phase plane (do the solutions go clockwise or counterclockwise around the origin?); and
(e) using HPGSystemsolver, sketch the x y -phase portrait and the $x(t)$ - and $y(t)$ graphs for the solutions with the indicated initial conditions.

5. $\frac{dY}{dt}=\left(\begin{array}{rr}\hfill -3& \hfill -5\\ \hfill 3& \hfill 1\end{array}\right)Y,$ with initial condition $Y_0=(4,0)$