Shown below is an electrical$-$mechanical system used for stamping operations in a$.$

manufacturing facility. The electrical system is comprised of a voltage source, switch,

resistor and inductor. The electrical system creates a force that is applied to the mechanical

system. The mechanical system is comprised of a mass, spring, and damper. Collectively

these elements create a solenoid. This force is a function of current within the circuit,

namely $F_{\text{s}\text{o}\text{l}}=i,$ and the back electromotive force $($EMF$)$ is $e_m=x^{}.$ Determine the

input$-$output relationship between the voltage input $e_m$ to the output displacement $x(t)$

when the switch closes.