# Differential equation question

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)

• what have you done? – mookid Apr 1 '14 at 18:02
• I have done all parts of the question except i do not know how to sketch the phase plane diagram for this equation. – jesse Apr 1 '14 at 18:03
• can you add your contribution to the question? – mookid Apr 1 '14 at 18:04
• Equilibrium: x= -2, 0, and 3 – jesse Apr 1 '14 at 18:07
• Edit the question. Then what have you done concerning stability? – mookid Apr 1 '14 at 18:09

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:

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