Question

Consider the following autonomous first order differential equation.

dy/dx=(−(y−4)2)(y+9)
The critical points are y=4 and y=−9. Classify these critical points (in the given order) as asymptotically stable, unstable, or semi-stable.
(A) semi-stable and unstable
(B) asymptotically stable and asymptotically stable
(C) asymptotically stable and unstable
(D) asymptotically stable and semi-stable
(E) unstable and asymptotically stable
(F) unstable and semi-stable
(G) semi-stable and semi-stable
(H) semi-stable and asymptotically stable
(I) unstable and unstable

Answer 1

Answer: (D) asymptotically stable and semi-stable

Explanation:

Given that:

dy/dx=(−(y−4)2)(y+9)

The critical points are y=4 and y=−9

We have to check the sign of dy/dx in the intervals (-inf,-9), (-9,4), (4, infinity)

In (-inf, -9), dy/dx , let's take y = -10, so we have

(-(-10-4)^2(-10+9) > 0

In (-9,4), let's take y = 0

dy/dx = (-(0-4)^2)(0+9) < 0

In (4, inf), let's take y = 10

So we have (-(10-4^)2(10+9) < 0

So at y = -9, sign change for dy/dx from positive to negative. SO it is asymptotically stable

And at y = 4, sign is negative on both sides. So 4 is semi stable