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6. Answer the questions below for the following system: or 3 5 w ( 5 _3 )Y (i) Calculate the eigenvalues and corresponding eigenveetors of the linear system. (ii) What type of equilibrium is the origin? Center, spiral sink, or spiral source? (iii) What is the natural frequency of the oscillations in the system? What is the natural period length? (iv) If YOU) = {1,1} then what is the derivative Y'{0)? Is the system spiral clockwise or counter-clockwise? (V) Find two linearly independent solutions Ym[t) and lim(t) such that the general solution of the system is of the form Y(t) = hIYm(t) l hail-t). (Vi) Find the partieular solution given by the initial condition YOU) = (1,1). (Vii) Using the formula. you derived in part (vi), plot the :r:(t) and y(t) curves on the same graph for f] 5 t 5 3. Make sure to indicate whieh curve is :r(t) and which eurve is y(t). (Hint: Yen may need to use the \". *\" symbol to per-fem element-wise multiplication of neetem. For example, to plot the function :r:(t) = e'm'sinHt), yen eenld use the code below) t_nin = 0; t_max = 3; t_step = .01; time = t_min:t_step:t_max; x = expC-2* time}.*sin{4 * time); plot{time,x); (viii) On the :r x yplane, plot the direction eld of our linear system. On top of your direction eld, plot the solution curve Y[t) = (a:[t),y(t}) you found in part [vi]. The dimensions of your graph should he [2, 2] X [2, 2] and you should plot the curve Y[t) for t6 [0,3]