# Find the critical points and sketch the phase portrait of

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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eninsala06
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:
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William Appel
You helped me at the most important moment, thanks