Problem #$1$: $(7$pts$)$

The following figure shows a translational mechanical network. $x_1$ and $x_2$ are the displacements of $M_1=5$ and $M_2=7$ masses respectively with independent driving force of magnitude $f(t)=9t^2$ on $M_2$ mass.

a$)$ Draw free body diagrams of the given masses.

b$)$ Write LTI ordinary differential equations for the given masses.

c$)$ Write Laplace transform equations of these system ODEs.

$($The quastion from Control System course$)$Please solve the quastion by your hand $)$