Consider the two-tank liquid-level system shown in Figure 7.17. The liquid is pumped into tanks 1 and 2 through valves of linear resistances \(R_{1}\) and \(R_{2}\), respectively. The pressure of the fluid increases by \(\Delta p\) when crossing the pump. The cross-sectional areas of the two tanks are \(A_{1}\) and \(A_{2}\), respectively. The liquid flows from tank 1 to tank 2 through a valve of linear resistance \(R_{3}\) and leaves tank 2

through a valve of linear resistance \(R_{4}\). The density \(ho\) of the liquid is constant. Derive the differential equations in terms of the liquid heights \(h_{1}\) and \(h_{2}\). Write the equations in second-order matrix form.

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FIGURE 7.17 Problem 5. Pa R