In “Analyze This!“, we examined system

*qualitatively*.

Now, let us seek its equilibrium *quantitatively*.

In theory, one may first solve *differential equation*

for , using a popular symbolic differential equation solver such as ‘ode2’. Then compute as , followed by .

However,in practice, such attempt meets a deadend rather quickly (see Fig. 1).

Fig. 1

Bring in a more sophisticated solver is to no avail (see Fig. 2)

Fig. 2

An alternative is getting directly from *polynormial* *equation*

We can solve (1) for if . For example, when , has three roots (see Fig. 3).

Fig. 3

First two roots are complex numbers. By Descartes’ rule of signs, the third root

is the of equilibrium (see *Exercise-1*).

*Exercise-1* Show the third root is real and positive.

*Exercise-2* Obtain from