$|$ A simple mechanical system is shown in the figure below. The parameters are $\backslash ($ k$=\backslash )$ spring constant, $\backslash ($ b$=\backslash )$ viscous friction constant and $\backslash ($ m$=\backslash )$ mass. A step of $4$ Newtons force $\backslash (($F$($s$)=4/$ s$)\backslash )$ is applied and the resulting step response is shown below.

a$)(2$ points$)$ Write the differential equation of motion of the mass in this system. Write the transfer function from the force $\backslash ($ F $\backslash )$ to the position $\backslash ($ x $\backslash )$ of the mass.

b$)(8$ points$)$ Based on the response of the system in the experiment described above, as shown in the plot, estimate the values of the system parameters $\backslash ($ k$,$ b $\backslash ),$ and $\backslash ($ m $\backslash ).$ Important notes: $(1)$ the magnitude of the step input force is not $1$ Newton; $(2)$ the peak output can be read as $0\mathrm{.}113,$ the peak time is $2$ seconds, and the steady$-$state output is $0\mathrm{.}1.$

Hint: measure the important performance values of the step response $($overshoot$,$ peak time, DC gain$),$ then use the formulae of these values to calculate the system parameters.