Problem $5.$ Consider the mechanical system

$m{q}_0^{}+c_2(q_0^{}-q_2^{})+k_2(q_0-q_1)=u$

$c_1{q}_1^{}+k_2(q_1-q_0)=0$

$c_2(q_2^{}-q_0^{})+k_1{q}_2=0$

where $u$ is the input and $y=q_0$ is the output. Note that these are the differential equations

from the system in Problem $9$ of Homework $1.$ Complete parts a$)$ and b$).$

a$)$ Write $(5)-(7)$ as an input$-$output differential equation.

b$)$ Write $(5)-(7)$ and $y=q_0$ in state$-$space form, where the state is $x_1=q_0,x_2=q_0^{},$

$x_3=q_1,$ and $x_4=q_2.$