Problem $3$: Textbook Problem $8\mathrm{.}21($Do parts b and c only$)$;

Note: To solve parts b and c $,$ use the modeling equations given in part $($a$).$

$8\mathrm{.}21.$

a$.$ For the translational mechanical system described in Problem $2\mathrm{.}4,$ let $M_1=M_2=B_1=B_2=B_3=K_1=K_2=K_3=1$ in a consistent set of units. Verify that the system can be modeled by the pair of differential equations

$x_1^{}+2x_1^{}+2x_1-x_2^{}-x_2=f_a(t)$

$-x_1^{}-x_1+x_2^{}+2x_2^{}+2x_2=0$

b$.$ Find the transfer function when the output is $x_1.$

c$.$ Repeat part $($b$)$ when the output is $x_2.$ Comment on the similarities and dissimilarities between the two transfer functions.