Find the critical points and sketch the phase portrait of

Quentin Johnson 2022-01-21 Answered
Find the critical points and sketch the phase portrait of the given autonomous first order differential equation. Classify the each critical point as asmyptotically stable, unstable, or semi-stable.
y=y2(4y2)
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eninsala06
Answered 2022-01-21 Author has 37 answers
Consider the provided autonomous first order differential equation.
y=y2(4y2)
The provided autonomous first order differential equation can be written as:
dydx=f(y)=y2(4y2)
We can get the critical points by solving f(y) = 0
y2(4y2)=0
y=0ory=±2
Therefore, the critical points are y = -2, 0 and 2.
Now to find the asymptotically stable, unstable or semi-stable as shown below:
y2(4y2)>0  when  4y2>0
Therefore,
y2<4
2<y<2
Thus, the f(y) is increasing in (-2,2)
Now,
y2(4y2)<0  when  4y2<0
Therefore,
y2>4
y<2 or y>2
(,2)(2,) the f(y) is decreasing
Now classify them as shown below:
at y=0, semi-stable {above increasingbelow increasing
at y=2, asymptoti stable {above increasingbelow increasing
at y=2, ustable {above increasingbelow increasing
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William Appel
Answered 2022-01-22 Author has 44 answers
You helped me at the most important moment, thanks
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