Equilibrium solution

Definition

An equilibrium solution of a differential equation is a constant solution where the derivative is zero, meaning the system is in a steady state. It represents a point where there are no changes over time.

5 Must Know Facts For Your Next Test

1. Equilibrium solutions are found by setting the derivative equal to zero and solving for the dependent variable.
2. In direction fields, equilibrium solutions appear as horizontal lines where all slope marks are level.
3. Stability of an equilibrium solution can be analyzed using phase plane analysis or eigenvalues in linear systems.
4. If perturbations from an equilibrium lead back to it, it is considered stable; if they lead away, it is unstable.
5. Nonlinear systems can have multiple equilibrium solutions, each with its own stability characteristics.

Review Questions

Related terms

Direction Field: A graphical representation showing the slopes of a differential equation at various points in the plane.

Phase Plane: A visual representation of trajectories of a dynamical system in the state-space.

Stability:

\text{The characteristic of an equilibrium solution that describes whether small perturbations return to or diverge from it.}