Equilibrium points that are not the origin Problem set A

Find the equilibrium point of each system.

1. (a) \(\dfrac{dx}{dt}=2x − y − 5,\quad \dfrac{dy}{dt}=x + 3y − 6\)
2. (b) \(\dfrac{dx}{dt}=−3x + y − 5,\quad \dfrac{dy}{dt}=2x − 4y\)

A system is \(\dfrac{dx}{dt}=−2x + y + 5,\quad \dfrac{dy}{dt}=x − 2y + 2\).

1. (a) Find the equilibrium point.
2. (b) Find the eigenvalues of the matrix of the system.
3. (c) State the kind of equilibrium point. What happens to \(x\) and \(y\) in the long run?

Two species of fish compete for food in a lake. Their numbers, in thousands, after \(t\) years satisfy \(\dfrac{dx}{dt}=x(10-x-y)\) and \(\dfrac{dy}{dt}=y(12-2x-y)\).

1. (a) Show that \((2, 8)\) is an equilibrium point.
2. (b) Find the other equilibrium points and say what each one means.

The numbers of hares \(x\) and foxes \(y\) on an island, in hundreds, after \(t\) years satisfy \(\dfrac{dx}{dt}=0.5x-0.02xy\) and \(\dfrac{dy}{dt}=0.004xy-0.3y\).

1. (a) Find the equilibrium points.
2. (b) Say what each equilibrium point means for the two populations.
3. (c) Start with \(x=80\) and \(y=25\). Use Euler's method with step length 0.1 to find \(x\) and \(y\) after one step. Is this start an equilibrium?

Two vats in a dye works are joined by a pipe. After \(t\) hours the vats hold \(x\) kg and \(y\) kg of dye, and satisfy \(\dfrac{dx}{dt}=−0.6x + 0.2y + 10,\quad \dfrac{dy}{dt}=0.3x − 0.5y + 11\).

1. (a) Find the equilibrium point.
2. (b) Find the eigenvalues of the matrix of the system and say what kind of equilibrium point it is.
3. (c) State the long-run amount in each, in kg.

A system is \(\dfrac{dx}{dt}=3x + y − 5,\quad \dfrac{dy}{dt}=2x + 2y − 6\).

1. (a) Find the equilibrium point.
2. (b) Find the eigenvalues and eigenvectors of the matrix of the system.
3. (c) Say what kind of equilibrium point it is and describe a sketch of the phase portrait: the lines to draw, the arrows and the curves.

A system is \(\dfrac{dx}{dt}=2x + y − 7,\quad \dfrac{dy}{dt}=kx + y − 5\), where \(k\) is a constant.

1. (a) For \(k=0\), find the equilibrium point.
2. (b) Find the values of \(k\) for which the equilibrium point is a saddle point.
3. (c) Show that there is no equilibrium point when \(k=2\).