(a) Sketch a phase portrait for the following system showing all isoclines:

dx dt = 3x + 2y, dy dt = x − 2y.

[6]

(b) Show that the system dx dt = xy − x2y + y3, dy dt = y2 + x3 − xy2 can be transformed into dr dt = r2 sin(θ), dθ

dt = r2(cos(θ) − sin(θ))(cos(θ) + sin(θ))

using the relations r ˙r = x ˙ x + y ˙ y and r2 ˙ θ = x ˙ y − y ˙ x. Sketch a phase portrait for this system given that there is one nonhyperbolic critical point at the origin.

[14]

422 18. Examination-Type Questions