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 $z(t)=y(t+t_{0})$ for any constant $t_{0}$

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

Solution

Autonomous equations can be solved by Separation of Variables method.

Phase Line

Consider $y'=F(y)$

• If $F(y)=0$, the solution is at equilibrium
• If $F(y)>0$, then y is increasing in t
• If $F(y)<0$, then y is decreasing in t