$(20$ points$)$ Consider the following $2$ degree of freedom system. Assume mass $1$ slides with no friction, and the cord suspending mass $2$ does not slip on the massless pulley wheel. Directions of positive motion are defined in red in the figure. Assume all coordinates $=0$ when springs are unstretched.

a$.(3)$ Write a kinematic constraint relating position of mass $1($x$\_1)$ to position of mass $2($y$\_2).$

b$.(7)$ Draw free body diagrams for all $3$ bodies in the system

c$.(10)$ Find equations of motion for this system, and report in matrix form. Use x$\_1$ and y$\_3$ as your variable coordinates. Note:y$\_2$ should not appear in your answer, due to your kinematic constraint from part $($a$).$