Question

Consider the following autonomous first order differential equation.

Answer 1

Answer: (D) asymptotically stable and semi-stable

Step-by-step 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