Find the equilibrium position of an object which satisfies the equation of motion Find the equilibrium position of an object which satisfies the equation of motion. I know how that there is a formula $$ d^{2}x/dt^{2} = - \omega^{2}x $$ but I can't see the usage of it when there are so many "things" on the right side of the equation.
$$ 4d^{2}x / dt^{2} = −x^{3} + x^{2} − x + 1 $$ 

Recall Newton's law
$$
\frac{{\rm d}^2 x}{{\rm d}t^2} = -\frac{{\rm d}V}{{\rm d}x}
$$
In you case, it is pretty straightforward to find the potential $V$
$$
V(x) = \frac{x^4}{16} - \frac{x^3}{12} + \frac{x^2}{8} - \frac{x}{4}
$$
You see that the dominant term is $\sim x^4$, this means that the potential goes to infinite, therefore, it must have an equilibrium point somewhere, to find it just calculate the locations where 
$$
\frac{{\rm d}V}{{\rm d}x} = 0
$$
If you do the math, you'll find the only real root is $x=1$ (red point in the graph)

