Find and classify the equilibrium solutions of the subsequent differential equation.

y' = y² - y - 6

Solution

The equilibrium solutions are to such differential equation are y = -2, y = 2, and y = -1. There is the sketch of the integral curves as:

From above it is clear ash hopefully that y = 2 is an unstable equilibrium solution as well as y = -2 is an asymptotically stable equilibrium solution. Though, y = -1 behaves differently from either of these two. Solutions which start above it move indirections of y = -1 whereas solutions that begin below y = -1 move away as t raises.

In cases where solutions on individual side of an equilibrium solution move in the directions of the equilibrium solution and on the other side of the equilibrium solution back off from it we say this as the equilibrium solution semi-stable.

Thus, y = -1 is a semi-stable equilibrium solution.