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Consider the differential equation $\dfrac{dx}{dt}=x^3−x^2−6x$ .

Find all equilibria.

Determine the stability of each equilibrium analytically (not from the phase line diagram).

Sketch the phase-line diagram.

I have solved--- $x=-2$ (unstable), $0$ (stable), and $3$ (unstable)

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  • $\begingroup$ what have you done? $\endgroup$ – mookid Apr 1 '14 at 18:02
  • $\begingroup$ I have done all parts of the question except i do not know how to sketch the phase plane diagram for this equation. $\endgroup$ – jesse Apr 1 '14 at 18:03
  • $\begingroup$ can you add your contribution to the question? $\endgroup$ – mookid Apr 1 '14 at 18:04
  • $\begingroup$ Equilibrium: x= -2, 0, and 3 $\endgroup$ – jesse Apr 1 '14 at 18:07
  • $\begingroup$ Edit the question. Then what have you done concerning stability? $\endgroup$ – mookid Apr 1 '14 at 18:09
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I see you showed two parts of the problem solution in the comments and it is important to show your work!

The equilibrium are found by solving:

$$x' = x^3 - x^2 - 6x = 0 \implies x_{1,2,3} = -2, 0, 3$$

To analytically test stability, we evaluate the derivative of $f(x) = x^3 - x^2 - 6x$ at each equilibrium point and this yields:

  • $f'(x) = 3x^2 - 2x - 6$
  • $f'(-2) = 10 \gt 0 \implies$ unstable
  • $f'(0) = -6 \lt 0 \implies$ stable
  • $f'(3) = 15 \gt 0 \implies$ unstable

A phase line is:

$~~~~~~~~~~~~~~$enter image description here

Compare the phase line drawing to a direction field plot and what do you notice?

enter image description here

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