 # 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.
${y}^{\prime }={y}^{2}\left(4-{y}^{2}\right)$
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Consider the provided autonomous first order differential equation.
${y}^{\prime }={y}^{2}\left(4-{y}^{2}\right)$
The provided autonomous first order differential equation can be written as:
$\frac{dy}{dx}=f\left(y\right)={y}^{2}\left(4-{y}^{2}\right)$
We can get the critical points by solving f(y) = 0
${y}^{2}\left(4-{y}^{2}\right)=0$
$y=0\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}y=±2$
Therefore, the critical points are y = -2, 0 and 2.
Now to find the asymptotically stable, unstable or semi-stable as shown below:

Therefore,
${y}^{2}<4$
$-2
Thus, the f(y) is increasing in (-2,2)
Now,

Therefore,
${y}^{2}>4$

Now classify them as shown below:
at
at
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