Find the critical points and phase portrait of the differential equation below. Classify each critical point as stable, unstable, or semi-stable. By hand, sketch typical solution curves in the regions in the xy-plane determined by the graphs of the equilibrium solutions. Show work to verify your arrow directions in each interval. dy/dx = y4 - 6y3 +8y2

The critical points are 0,2,4

0 is semi-stable, 2 is stable, 4 is unstable.

Check the phase line and the solution curves in the files attached below

Step-by-step explanation:

To get the critical points, dy/dx = 0

To classify the stability, the interval is divided into

If y = -1, f(-1) = 15 > 0

If y = 1, f(1) = 3 > 0

If y = 3, f(3) = -9 < 0

If y = 5, f(5) = 75 > 0

The phase line is drawn in the file attached.

By critically observing the phase line:

The point y = 4 is unstable because the arrows are moving away from it

The point y = 2 is stable because the arrows are moving towards it

The point y = 0 is semi-stable because one arrow is moving towards it while the other moves away from it.

The typical solution curves are sketched in the file attached below.

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