Problem #$3.(30$ points$).$ Figure #$2$ below shows a translational mechanical system where external force $u(t)$ is the input to the system, and the displacement $y(t)$ of the mass $(m)$ is the output. The damper constant is $b$ and spring constant is $k.$

$(15$ points$).$ Draw the free$-$body diagram and write the ordinary differential equation $($ODE$)$ of the dynamic system assuming that the initial conditions are zero.

$(15$ points$).$ Find the transfer function $\frac{Y(s)}{U(s)}$ of the translational mechanical system where $y(t)$ is the displacement of the mass and $u(t)$ is the external force applied to the system.

Figure #$2.$ Translational Mechanical System.