Autonomous ODE: Difference between revisions
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(Created page with "'''Autonomous ODE's''' have no explicit t-dependence. They come in the form <math> y' = F(y) </math> = Equilibrium = Autonomous ODE's have trivial ODE solutions. If <math> F(c) = 0 </math> then <math> y(t) = c </math> is the equilibrium solution of the ODE. If <math>y(t)</math> is a solution, then so is <math>z(t) = y(t + t_0)</math> for any constant <math>t_0</math> <math> \begin{aligned} y'(t) &= F(y(t))\\ z'(t) &= y'(t + t_0) \\ &= F(y(t + t_0)) \\ &= F(z(t)...") |
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</math> | </math> | ||
= Equilibrium = | = Equilibrium Solutions = | ||
Autonomous ODE's have trivial ODE solutions. | Autonomous ODE's have trivial ODE solutions. | ||
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</math> | </math> | ||
is | is an equilibrium solution of the ODE. | ||
If <math>y(t)</math> is a solution, then so is <math>z(t) = y(t + t_0)</math> for any constant <math>t_0</math> | If <math>y(t)</math> is a solution, then so is <math>z(t) = y(t + t_0)</math> for any constant <math>t_0</math> | ||
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</math> | </math> | ||
= Solution = | = General Solution = | ||
Autonomous equations can be solved by [[Separation of Variables]] method. | Autonomous equations can be solved by [[Separation of Variables]] method. | ||
= | = Equilibrium Analysis = | ||
Consider <math>y' = F(y)</math> | Consider <math>y' = F(y)</math> | ||
* If <math>F(y) = 0</math>, the solution is at equilibrium | * If <math>F(y) = 0</math>, the solution is at ''equilibrium'' | ||
* If <math>F(y) >0</math>, then ''y'' is increasing in ''t'' | * If <math>F(y) >0</math>, then ''y'' is increasing in ''t'' | ||
* If <math>F(y) < 0</math>, then ''y'' is decreasing in ''t'' | * If <math>F(y) < 0</math>, 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. | |||
Revision as of 22:28, 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.
