P$2.(30$ points$)$ In the figure before, a motor drives a pump to supply a hydraulic cylinder. The DC motor produces a torque $\backslash ($ T$\_\{$p$\}\backslash )$ that accelerates the pump inertia $\backslash ($ J$\_\{$p$\}\backslash ).$ You may assume that internal pump friction losses may be modeled as a damping term $\backslash ($ b$\_\{$p$\}\backslash ).$ Assume that the pump has a leakage resistance $\backslash ($ R$\_\{\backslash$text $\{$leak $\}\}\backslash ).$ The pump has displacement $\backslash ($ D$\_\{$p$\}\backslash )$ and produces a flow that lifts a mass $\backslash ($ m $\backslash ).$ Assume a fluid density of $\backslash (\backslash$rho $\backslash ).$

Figure Prob. $2$: Mechanical$/$fluid system $($both pictures depict the same system$)$

a$)$ Write the resulting system of differential equations. $($Hint: you will end up with two differential equations.$)$

b$)$ Determine the system transfer function from $\backslash ($ T$\_\{$p$\}\backslash )$ to $\backslash ($ x $\backslash )($assume zero initial condition$).$