$(20$ pts$)$ The system shown in the figure below is used to move the spool valve of a hydraulic

servomotor. Both the spool valve and piston are assumed to be massless, and there are no leakages

in the cylinder. Furthermore, let us assume that the spool valve obeys the underlapped valve model

$q=C_{1}x-C_{2}P,$ where $C_1$ and $C_2$ are constants and $P=P_{in}-P_{\text{o}\text{u}\text{t}}.$

a$.(10$ pts$)$ Develop a model of the system with the spool valve force $f(t)$ as the input and the

load displacement $y(t)$ as the output.

b$.(10$ pts$)$ Assuming the spool valve damping $B_1$ is negligible, combine the equations developed

in part $($a$)$ into one ODE relating the load displacement $y(t)$ to the spool valve force $f(t).$