Problem $1(600pts)$ : For the two$-$mass system shown in figure $1,$ derive the

model of the system $($i$.$e$.,$ differential equations describing the dynamics of the

system$)$ and then find the transfer function for the following cases:

a$)u(t)$ is input and $x(t)$ is output

b$)u(t)$ is input and $y(t)$ is output

Figure $1$: two$-$mass system

Problem $2(400pts)$ : Derive a state variable model of Problem $1,$ consider the vector $y(t)=[x(t)y(t){]}^T$

as the output of the system $($i$.$e$.,$ the system is single input$-$multiple outputs$).$