Problem $3(30$ points$).$

You are given the two$-$cart system below. Mass of the carts, spring constants and damping coefficients

are shown. There are two forces acting on the carts as shown. Using Newtonian laws of motion find the

differential equations that describe the motion of the carts. Represent it in a linear state$-$space form x$^()=$

Ax$+$Bu$.$ Be careful, there is a catch!