equilibrium points of an differential equation

Question

I want to find the equilibrium points of this differential equation:

$y'(t)=ay(1-y)-by$

i have found that $y=\frac{(a-b)}{a }$makes $y'(t)=0$ but i don't know if $y'(t)=0$ is also an equilibrium solution

thanks for the help!

Answer 1

Any value of $y$ that makes $y'=0$ is an equilibrium point. If $ay(1-y)-by=0$ then either $y=0$ or $a(1-y)-b=0$. The latter implies $y=(a-b)/a$. So $0$ and $(a-b)/a$ are both equilibrium points, and there are no others.

Answer 2

An equilibrium solution is a constant solution to a differential equation. If you draw a slope field, the equilibrium solution is a horizontal line . So if you'd like to find the equilibrium solution for an OE, you have to put the OE equal to zero and solving for the variable value. Exactly what @Micheal did in details.