Phase portraits for real eigenvalues Problem set A

Use the characteristic equation \(\det(\mathbf M-\lambda\mathbf I)=0\).

1. (a) Find the eigenvalues of \(\mathbf M=\begin{pmatrix}7&2\\-2&2\end{pmatrix}\).
2. (b) Find the eigenvalues of \(\mathbf M=\begin{pmatrix}-4&2\\3&-5\end{pmatrix}\).

For the system \(\dfrac{dx}{dt}=4x+y\) and \(\dfrac{dy}{dt}=2x+3y\), find the eigenvalues and eigenvectors of the matrix of the system, say what kind of equilibrium point the origin is, and sketch the phase portrait. Show the eigenvector lines and the direction of the arrows.

For the system \(\dfrac{dx}{dt}=-x+y\) and \(\dfrac{dy}{dt}=-3x-5y\), find the eigenvalues and eigenvectors of the matrix of the system, say what kind of equilibrium point the origin is, and sketch the phase portrait. Show the eigenvector lines and the direction of the arrows.

For the system \(\dfrac{dx}{dt}=2x+y\) and \(\dfrac{dy}{dt}=4x-y\), find the eigenvalues and eigenvectors of the matrix of the system, say what kind of equilibrium point the origin is, and sketch the phase portrait. Show the eigenvector lines and the direction of the arrows.

Two lakes are joined by a channel. After \(t\) years the amounts of a pollutant in the two lakes are \(x\) tonnes and \(y\) tonnes, modelled by \(\dfrac{dx}{dt}=-4x+y\) and \(\dfrac{dy}{dt}=2x-3y\).

1. (a) Find the eigenvalues and eigenvectors of the matrix of the system, and classify the origin.
2. (b) Describe the phase portrait: the eigenvector lines, the arrows and the shape of the curves.
3. (c) At the start \(x=3\) and \(y=6\). Show that this point lies on an eigenvector line and describe how the two amounts change.

A toy boat floats on a stream. Its position relative to a post at the origin, \(t\) seconds after release, is \((x,y)\) metres, modelled by \(\dfrac{dx}{dt}=-x+2y\) and \(\dfrac{dy}{dt}=3x\).

1. (a) Find the eigenvalues and eigenvectors of the matrix of the system, and classify the origin.
2. (b) The boat is released at \((2,-2)\). Describe its path.
3. (c) The boat is released at \((1,1)\) instead. Describe what happens in the long run.

Consider the system \(\dfrac{dx}{dt}=3x+ky\) and \(\dfrac{dy}{dt}=2x+y\), where \(k\) is a constant.

1. (a) Find, in terms of \(k\), the product of the eigenvalues of the matrix of the system.
2. (b) Hence find the values of \(k\) for which the origin is a saddle point.
3. (c) For \(k=0\), find the eigenvalues and classify the origin.