Differential equations

The differential equations can be used to describe the behavior of a system over time. Hereto, the changes in time of a system are prescribed as a function of the current state of the system.

For example, the dynamics of a mechanical system with mass

(5.1) |

where

Another possible notation is the following form

(5.2) |

where

Hence, since

To obtain a trajectory satisfying the system equations, an initial condition should be specified.
When an initial condition is specified, one can look for a trajectory with this initial condition that satisfies the differential equation. For example, in the above example, a system with initial position

(5.3) |

Plotting this function in time yields the trajectory depicted in Figure 5.2.

This trajectory can also be plotted in the phase plane, such that one obtains Figure 5.3.

As can be seen in this figure, the vector field is always locally tangent to the trajectories

Esteur 2010-03-22