For the system shown below, the angular motion of the ideal lever from the vertical position is small, so the motion of the top and midpoint can be regarded as horizontal. The input is the force $\backslash ($ f$\_\{$a$\}($t$)\backslash )$ applied at the top of the lever, and the output is the support force on the lever, taking the positive sense to the right.

a$.$ Select a suitable set of state variables and write the corresponding state$-$variable model.

b$.$ Find the input$-$output differential equation relating $\backslash ($ x $\backslash )$ and $\backslash ($ f$\_\{$a$\}($t$)\backslash ).$