Autonomous ODE

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

$y'=F(y)$

Equilibrium

Autonomous ODE's have trivial ODE solutions.

If

$F(c)=0$

then

$y(t)=c$

is the equilibrium solution of the ODE.

If $y(t)$ is a solution, then so is Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle z(t)=y(t+t_{0})} for any constant $t_{0}$

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}y'(t)&=F(y(t))\\z'(t)&=y'(t+t_{0})\\&=F(y(t+t_{0}))\\&=F(z(t))\end{aligned}}}

Solution

Autonomous equations can be solved by Separation of Variables method.

Phase Line

Consider $y'=F(y)$

• If Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F(y)=0} , the solution is at equilibrium
• If Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F(y)>0} , then y is increasing in t
• If Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F(y)<0} , then y is decreasing in t