Autonomous ODE: Difference between revisions

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This can be visualized on a '''phase line'''.
This can be visualized on a '''phase line'''.


Some equilibrium solutions are ''stable'', where the solutions converge and slight perturbations in ''y'' will not result in drastic changes in the solution. In contrast, some other equilibrium solutions are ''unstable'', where slight perturbation will result in drastic changes.
Some equilibrium solutions are [[Equilibrium#Stability|stable]], where the solutions converge and slight perturbations in ''y'' will not result in drastic changes in the solution. In contrast, some other equilibrium solutions are ''unstable'', where slight perturbation will result in drastic changes.

Revision as of 23:14, 15 April 2024

Autonomous ODE's have no explicit t-dependence. They come in the form

Equilibrium Solutions

Autonomous ODE's have trivial ODE solutions.

If

then

is an equilibrium solution of the ODE.

If is a solution, then so is for any constant

General Solution

Autonomous equations can be solved by Separation of Variables method.

Equilibrium Analysis

Consider

  • If , the solution is at equilibrium
  • If , then y is increasing in t
  • If , then y is decreasing in t

This can be visualized on a phase line.

Some equilibrium solutions are stable, where the solutions converge and slight perturbations in y will not result in drastic changes in the solution. In contrast, some other equilibrium solutions are unstable, where slight perturbation will result in drastic changes.