Question

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

Answer 1

Answer:

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:

$dy/dx = y^{4} - 6y^{3} + 8y^{2}$

To get the critical points, dy/dx = 0

$y^{4} - 6y^{3} + 8y^{2} = 0\\y^{2} ( y^{2} - 6y + 8) = 0\\y^{2} ( y-2)(y-4) = 0\\y = 0, 2, 4$

To classify the stability, the interval $(-\infty, \infty)$ is divided into $(-\infty, 0), (0,2), (2,4),(4, \infty)$

$f(y) = y^{4} - 6y^{3} +8y^{2}$

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.