Coupled systems and phase portraits Problem set B

Find the eigenvalues and eigenvectors of the matrix \(\begin{pmatrix}1&3\\4&2\end{pmatrix}\).

A system is \(\dot x=2x + 2y,\ \dot y=x + 3y\).

1. (a) Find the eigenvalues and eigenvectors of the matrix of the system.
2. (b) Sketch the phase portrait, showing the directions of the eigenvectors, and name the type of equilibrium.

Two species of plant are introduced to an island. Their numbers, in hundreds, are modelled by \(\dot x=3x + y,\ \dot y=2x + 4y\), where t is in years. At \(t=0\) there are 5 hundred of species X and 1 hundred of species Y.

1. (a) Find the eigenvalues and eigenvectors of the matrix of the system.
2. (b) Write down the general solution, then find x(t) and y(t) for the starting values given.
3. (c) Comment on how suitable the model is in the long term.

The numbers of wolves (x) and deer (y) in a park, in hundreds, are modelled by \(\dot x=7x - 10y,\ \dot y=5x - 8y\), where t is in years. Initially there are 17 hundred wolves and 10 hundred deer.

1. (a) Find x(t) and y(t).
2. (b) Show the solution curve on a phase portrait for \(0\le x,y\le50\). Show the directions of the two eigenvectors.
3. (c) What is the ratio of wolves to deer in the long term?

The deviations of the temperature (x, in degrees) and the pressure (y, in kPa) of a gas in a sealed tank from their set values are modelled by \(\dot x=-x + 4y,\ \dot y=-x - 3y\), where t is in minutes, with \(x(0)=-3\) and \(y(0)=4\).

1. (a) Use technology to sketch the solution curve in the x-y plane.
2. (b) Sketch the graphs of x and y against t for \(0\le t\le8\) on separate diagrams.
3. (c) Describe how x and y change with t.

The diagram shows the direction field for a system of differential equations in x and y, and the solution curve starting at \((10,6)\).

1. (a) Estimate the maximum value of y along the solution curve.
2. (b) Estimate the long-term state of the system.

The diagram shows the direction field for a system of differential equations in x and y. Say whether each statement is true or false.

1. (a) Whatever the starting values, x approaches 7 and y approaches 6.
2. (b) y always moves towards 6.
3. (c) If x is greater than 7, then x decreases.
4. (d) x always increases.