The mathematical model of a nonlinear system is provided. Using the procedure outlined in this section, find the operating point and derive the linearized model.

\(\ddot{x}+\dot{x}+x \sqrt{|x|}=\frac{1}{8}+\sin \left(\frac{1}{2} t\right), \quad x(0)=0, \dot{x}(0)=1\)