Figure 7.3 shows a pneumatic system, in which the pneumatic capacitances of the two rigid containers are \(C_{1}\) and \(C_{2}\), respectively. Dry air at a constant temperature passes through a valve of linear resistance \(R_{1}\) into container 1 . The pressure \(p_{\mathrm{i}}\) at the inlet of the valve is constant, and it is greater than \(p_{1}\). The air flows from container 1 to container 2 through a valve of linear resistance \(R_{2}\). Derive the differential equations in terms of the pressures \(p_{1}\) and \(p_{2}\). Write the equations in second-order matrix form.

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FIGURE 7.3 Problem 3. Pi P, C R Pz. Cz