Consider the system x = x2, y = xy, z = y xz,with initial conditions (x(0), y(0), z(0)) = (x0, y0, z0).

a) Find the general solution (x(t),y(t),z(t) with the given initial con-

ditions.

b) Find a conserved quantity F(x,y) which is a function of x and y

only. Verify that d F (x(t), y(t)) = 0. Hint: the problem is equivalent to find dt

the equation for the trajectory, as different trajetories are characterised by the relation F (x, y) = c1.

c) Using the relation F (x, y) = c1 obtained in b), eliminate either x or y (say y = F (x, c1) ) to reduce the system into one with only two variables x and c1. Find another conserved quantity G(x, z, c1) involving x and z, as well as the constant c1, hence a second conserved quantity G(x, z, F (x, y)) involving x, y and z.

(Notice that the solution for this system does not exist for all time, but we do not need to worry about it here. In fact, the system can be slightly modified while the answers to part b)and c) stay the same.)