$2\mathrm{.}19.$ The system shown in Figure $P2\mathrm{.}19$ has a nonlinear spring that obeys the expression $f_K=x^3.$

a$.$ Write the differential equation describing the system in terms of the displacement $x.$

b$.$ Let $x=x_0+z,$ where $x_0$ is the constant displacement caused by the gravitational force when the system is in static equilibrium. Rewrite the differential equation in terms of the variable $z,$ canceling the gravitational term.

c$.$ Comparison of $(21)$ and $(24)$ in Example $2\mathrm{.}5$ for a linear spring showed that the differential equation in $z$ was the same as the one in $x$ except for the deletion of the Mg term. Compare the results of parts $($a$)$ and $($b$)$ to show that such is not the case in this problem.