Consider the dynamical system
$$\dot x= cx - \frac{x}{1+x^{2}}$$ for $x\in\mathbb{R}$, with $c$ a positive constant.
Establish the location and number of equilibrium points of the system for all values of $c$. Then using linear stability analysis, determine the stability of the equilibrium points for all values of $c$.
My solution so far:
I set the RHS to $0$ and rearranged to get $$x(cx^2+(c-1))=0$$
so then $x=0$ or $cx^2+(c-1)=0$.
This second equation can be rearranged as $cx^2=1-c$
so when $c=1$ we have $x=0$, when $c\gt1$, we have no solutions.
But I am not sure what happens in the case $0\lt c \lt 1$.