Two tanks A and B each holding 40 L of the tears of differen

Two tanks A and B, each holding 40 L of the tears of differential equations students, are interconnected by pipes. The liquid flows from tank A into tank B at a rate of 4 L/min and from B into A at a rate of 1 L/min. The liquid inside each tank is kept well stirred. Pure water flows into tank A at a rate of 3 L/min, and the solution flows out of tank B at 3 L/min. If, initially, tank A contains 3.5 kg of salt and tank B contains no salt (only water), write a system of differential equations that model the mass of salt in each tank at time t greaterthanorequalto 0. Write the model in matrix form. Be sure to label your variables and include initial conditions. Use Maple or the DEExplorer app to plot the slope field of the system in the phase plane. Sketch or attach a printout with your exam. Is there an equilibrium solution? If so, where? Is it stable or unstable? Physically, what does the equilibrium solution mean?

Solution

Let start with generalize the differential equation

Consider a tank, which initially hold V₀ L of brine that contains a kg of salt. Another brine solution, containing salt b kg /L, is poured into the tank at the rate of e L/min, the well stirred solution leaves the tank at the rate of f L/min.

Let Q denote the amount ( in kg) of salt in the tank at any time. The time rate of change of Q, dQ/dt, equal the rate at which enters the tank minus the rate at which salt leaves the tank.

Salt enters the tank at the rate of be kg/min.

Volume of brine in the tank at any time t = V₀ + et + ft

The concentration of salt in the tank at any time = Q/( V₀ + et + ft)

Salt leaves the tank at the rate of f Q/( V₀ + et + ft) kg/min

    dQ/dt = be - f Q/( V₀ + et + ft)

dQ/dt + f Q/( V₀ + et + ft) = be , which is required generalized differential equation.

Now   V₀ = 40L ,     f = 3L/min,      e = 4L/min,     b=0

Thus             dQ/dt + 3Q/(40 + t) = 0

On solving       Q(40 + t)³ = c

Now at      t= 0 ,     Q=a = 3.5      c = 224000

         Q(40 + t)³ = 224000