The figure above shows a system which consist of a mass $($M$),$ spring $($k$),$ and two dampers $($b$1,$ b$2).$ The spring is attached at one end to the mass and to the damper $($b$2),$ which is massless at the other. The mass is pulled by a velocity source $($v$($t$)).$ x denotes the extension of the spring. v$1$ is the velocity of the mass and v$2$ is the velocity of the second damper. M $=1$ kg$,$ b$1=1$ Ns$/$m$,$b$2=9$Ns$/$m$,$k$=9$N$/$m$.$

a$)$ find v$1(0),$ v$2(0),$ and x$(0)$

b$)$ find x$(0+)$ and x $(0+)$

c$)$ write the differential equation for the system in terms of x$.$ d$)$ find x$($t$)$ given the initial conditions in c$.$