For the interconnected tanks problem of Section 5.1, page 241 , suppose that instead of pure water being fed into the tank A, a brine solution with concentration is used; all other data remain the same. Determine the mass of salt in each tank at time if the initial masses are and .
The change of mass of salt n tanks A and B are:
Let’s first derive a system of differential equations that describes the change of salt in each tank at a time t. One knows that and in the tank t, two pipes bring salt in it, the left one at the rate role="math" localid="1664171528030" , and the right lower pipe brings salt at the rate . The upper pipe carries salt out of the tank A at the rate of . So, one has that the change of the concentration of salt in the tank A is .
One has that the upper pipe brings salt into the tank B by the rate of , the lower pipe carries salt out of tank B by the rate and the right pipe carries salt out by the rate of so the change of concentration of salt in the tank B is
So, to determine the mass of salt in each rank one has to solve the following system:
The second equation gives us that
Substituting this into the first equation of the system one will get
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