pumpisopen toThe figure below shows a liquid-level system in which two tanks have cross-sectional areas, A₁ and A2, respec-tively. A pump is connected to tank 1 through a valve of linear resistance R₁. The inlet to theatmosphere, and the pressure of the fluid increases by Ap when crossing the pump. The liquid flows from tank 1 totank 2 through a valve of linear resistance R₂ and leaves tank 2 through a valve of linear resistance R3, exiting at at-mospheric pressure. Assume the density p of the liquid is constant and note that both tanks are open to atmosphereas shown.Pa ApYour Tasks:R₁PaÍRNA₁R₂PlePaA₂R3→ 9A. Derive a differential equation model for the system behavior in terms of the liquid heights h₁ and h₂.B. Put the differential equations into second-order matrix form.

To Find:A. The differential equation model for the system behavior in terms of the liquid heights and B. The second-order matrix form of the differential equation.Calculation:Mass balance for Tank 1:The rate of change of liquid in Tank 1 is given by the difference between the flow in and the flow out.The flow in is due to the pump, and the flow out is due to the valve connecting tank 1 to tank 2.Mass balance for Tank 2:The rate of change of liquid in tank 2 is given by the difference between the flow in from tank 1 and the flow out through the valve connected to tank 2Combine the equations to obtain the differential equation model for the system.the system of the differential equations can be expressed in matrix form. Where,X is the state vector.u is the input vector.A is the state matrix.B is the input matrix.Let be the state vector and be the input vector.The state matrix A and input matrix B can be extracted from the differential equations.Therefore, The second-order matrix is,This matrix form allows for a systematic analysis and control design for the liquid-level system.Thus,A. The differential equation model for the system behavior in terms of the liquid heights h1 and h2:B. The second-order matrix form of the differential equation: