Problem $3$: $(25$ points$)$ Carefully read sections $2\mathrm{.}1$ and $2\mathrm{.}2$ in the text. The mechanical system in the

figure below has as a single input, the external force $u(t),$ and two outputs $y_1(t)=x_1$ and $y_2(t)=x_3.$ The

friction force retarding $($preventing$)$ the motion of the mass $M_i$ is proportional to the mass velocity $y_i^{}$ with

a proportionality constant $B_i,$ for $i=1,2.$

a$)(6$ points$)$ Draw a free body diagram for each mass shown in the figure above.

b$)(6$ points$)$ Apply Newton's second law at each mass to obtain two, coupled, second$-$order ODEs in

terms of the displacements $x_1(t)$ and $x_3(t)$ and the input force $u(t).$

c$)(7$ points$)$ Using the two ODEs obtained in part $2,$ obtain an all$-$integrator block diagram representation

of the system. $\{$Hint: You will need two chains of integrators. One chain will take as an input $x_1^{}(t),$

while the other chain takes $x_3^{}(t)$ as an input.