Consider the differential equation below. (you do not need to solve this differential equation to answer this question.) y' = y^2(y + 4)^3 find the steady states and classify each as stable, semi-stable, or unstable. draw a plot showing some typical solutions. if y(0) = -2 what happens to the solution as time goes to infinity?

We have when or , so we need to check the sign of on 3 intervals:

Suppose . In particular, let . Then . Since is negative on this interval, we have as .Suppose , say . Then , so that as .Suppose , say . Then , so that as .

We can summarize this behavior as in the attached plot. The arrows on the -axis indicate the direction of the solution as . We then classify the solutions as follows.

is an unstable solution because on either side of , does not converge to the same value from both sides. is a semi-stable solution because for , solutions tend toward the line , while for solutions diverge to negative infinity.

Wow. that is a really cool question.