Autonomous ODE

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y' = F(y) }

They can be solved by Separation of variables method.

Equilibrium Solutions

Autonomous ODE's have trivial ODE solutions.

If

$F(c)=0$

then

$y(t)=c$

is an equilibrium solution of the ODE.

If $y(t)$ is a solution, then so is $z(t)=y(t+t_{0})$ for any constant $t_{0}$

${\begin{aligned}y'(t)&=F(y(t))\\z'(t)&=y'(t+t_{0})\\&=F(y(t+t_{0}))\\&=F(z(t))\end{aligned}}$

General Solution

Autonomous equations can be solved by Separation of Variables method.

Equilibrium Analysis

Consider Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y' = F(y)}

• If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(y) = 0} , the solution is at equilibrium
• If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(y) >0} , then y is increasing in t
• If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(y) < 0} , 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.