Write out the $3$ nonlinear differen$3$al equa$3$ons for the car$/$cylinder$/$spool system. These should be in terms of the spool orifice length $(!),$ the height of the car $(,$ which is measured from the sta$3$c equilibrium height of the car if there was no cylinder$),$ pressure in the upper cylinder compartment $("),$ and pressure in the lower cylinder compartment $($#$),.$ Keep everything symbolic. When considering the volumes of the upper and lower cylinder compartments, just assume they are constant and equal to the values given on the previous page $($this will not affect the accuracy of our simula$3$on very much because the fluid bulk modulus is so large, and because we will analyze this system within a rela$3$vely narrow region of opera$3$on$).$