# System of Linear DEs Imaginary Eigenvalues

- Author:
- Erik Jacobsen

- Topic:
- Differential Equation

Shown below is the phase portrait for a linear system of differential equations with constant coefficients and two imaginary eigenvalues. Here, the derivatives are with respect to time, and we are interested in the behavior of the solutions to the differential equation as time increases from zero.
Each solution to the differential equation is called a trajectory. The initial values (at

*t*= 0) of each solution are shown as points (red dots) in the phase plane (the xy-plane).Move the slider to adjust the real part of the eigenvalue for this system.
Try dragging the initial value points closer to and further from the origin.
Do the solutions move toward the origin or away from it? See if you can understand the names shown in the figure for various values of

*a*.