Question 1:

Consider the pair of differential equations

dx/dt= 1-y,

dy/dt= x^2 - y^2.

(a) Find all the equilibrium points of these equations.

(b) Classify each equilibrium point of this non-linear system as far as Possible by considering the Jacobian matrix.

Question 2:

Consider the pair of differential equations

dx/dt=(x^2 - 1)(y + 2),

dy/dt= x(y^2 - 1).

(a) Find all the equilibrium points of these equations.

(b) Using Maxima, or otherwise, plot the phase portrait of these equations.

(c) Briefly describe the phase path that goes through the origin.

Question 3:

Consider the pair of differential equations

dx/dt=(1 - 2x/π)cos y +(1 - 2y/π)sin x, dy/dt= sin x + cos y.

(a) Show that (π, 1/2π) is an equilibrium point of these equations.

(b) Classify this equilibrium point of this non-linear system as far as possible by considering the Jacobian matrix.