Chapter 3.6, Problem 10P

For each of the systems in Problem $7$ through $12$:

a) Find all of the critical points.

b) Find an equation of the form $H\left(x,y\right)=C$ satisfied by the solutions of the given system.

c) Using a computer, plot several level curves of the function $H$. These are trajectories of the given system. Indicate the direction of motion on each trajectory.

d) Describe behaviour of the trajectories near each critical point.

$\frac{dx}{dt}=-x+y+x^2,\frac{dy}{dt}=y-2xy$