For the system x˙ = −3x + y y˙ = x − 3y

(i) Show that points (x0, y0) = (4, 8) and (x0, y0) = (4, 2) remain in quadrant I, as in figure 4.9.

(ii) Show that points (x0, y0) = (−4, −8) and (x0, y0) = (−4, −2) remain in quadrant III, as in figure 4.9.

(iii) Show that points (x0, y0) = (2, 10) and (x0, y0) = (−2, −10) pass from one quadrant into another before converging on equilibrium.

(iv) Does the initial point (x0, y0) = (2, −5) have a trajectory which converges on the fixed point without passing into another quadrant?