Trajectories of Linear Systems

Part 1: Real Eigenvalues

In this module we study a number of systems of differential equations of the form

or, in matrix-vector notation,

where A is the matrix

In your worksheet you will find instructions for

• computing the eigenvalues and eigenvectors of the matrix A, and
• drawing trajectories on a direction field for the system.

For each of the following systems (specified by the matrix A):

• Record the eigenvalues and eigenvectors.
• Draw enough trajectories to get a comprehensive view of the phase portrait of the system. (You will have to add sets of initial conditions to those given.)
• Explain what features of the phase portrait correspond to values of the eigenvalues and directions determined by the eigenvectors.

1. A =
2. A =
3. A =
4. A =
5. A =