Question

dx/dt=y
dy/dt=x-3y

Answer 1

(a) Determine and plot the equilibrium points and nullclines for the system.
b) Show the direction of the vector field between the nullclines.
c) Sketch some solution curves starting near, but not on, the equilibrium points.
d) Label each equilibrium as stable or unstable and describe the long-term behavior.

Answer 2

If I remember it right, equilibrium points occur when both \(\dfrac{dx}{dt}\) and \(\dfrac{dy}{dt}\) are zero. The former occurs when \(y=0\), since \(\dfrac{dx}{dt}=y=0\), which means \(\dfrac{dy}{dt}=0\) when \(x\) is also 0.

Answer 3

I can't say I'm familiar with finding nullclines, so I can't really help you there.