How do you find a stationary (time independent) solution to a differential equation? What does it mean for a stationary solution to be stable or unstable and how do I know if it is stable or not?

Question

The equation differential equation given was dy/dt= 3y^2-9y, which describes the evolution of the population, with no initial conditions. It asks to find any stationary (time independent) solutions and which ones are stable and which are unstable. How do I find this? How can I find out if the solutions are stable or not? What does it mean for a stationary solution to be stable or unstable?

Answer 1

For an autonomous first-order equation [dy/dt = f(y)], your stationary solutions [y(t) = constant] are found by setting dy/dt = 0 and thus solving f(y) = 0. In this example, you'll have two of them.

If y(t) = c is any stationary solution, the value of f'(y) often will tell you about stability. If f'(c) > 0, the solution is unstable. If f'(c) < 0, the solution is stable (more precisely, "asymptotically stable"). If f'(c) = 0, you'll need a different method to determine stability [graphing f(y) versus y and looking for increasing/decreasing information].

Informally, "unstable" means that there are solutions that start arbitrarily close to the stationary solution, but move further away as time increases. "Asymptotically stable" means that solutions that start close enough to the stationary solution will approach the stationary solution as t approaches infinity.