Problem $4.$ Consider the mechanical system

$m_1{q}_1^{}+c_1{q}_1^{}+k_1(q_1-q_2)=u$

$m_2{q}_2^{}+c_2{q}_2^{}+k_2(q_2-q_1)=0$

where $u$ is the input and $y=q_1-q_2$ is the output. Complete parts a$)$ and b $).$

a$)$ Write $(3)$ and $(4)$ as an input$-$output differential equation.

b$)$ Write $(3)$ and $(4)$ and $y=q_1-q_2$ in state$-$space form, where the state is $x_1=q_1,$

$x_2=q_1^{},x_3=q_2,$ and $x_4=q_2^{}.$