$2\mathrm{.}21.($a$)$ Write a differential equation model for the system shown in Figure P$2\mathrm{.}21.$

$($b$)$ Derive the transfer function $G_1(s)=x_1\frac{s}{F}(s).$

$($c$)$ Derive the transfer function $G_2(s)=x_2\frac{s}{F}(s)$

FIGURE P$2\mathrm{.}21$