The equation of motion of a spring-mass-damper system, with a softening-type spring, is given by (in SI units)

\[100 \ddot{x}+500 \dot{x}+10,000 x-400 x^{3}=0\]

a. Determine the static equilibrium position of the system.

b. Derive the linearized equation of motion for small displacements \((x)\) about the static equilibrium position.

c. Find the natural frequency of vibration of the system for small displacements.