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) }

Equilibrium Solutions

Autonomous ODE's have trivial ODE solutions.

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(c) = 0 }

then

$y(t)=c$

is an equilibrium solution of the ODE.

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 y(t)} is a solution, then so is 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 z(t) = y(t + t_0)} for any constant 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 t_0}

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 \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.