Two tanks A and B, each of volume V , are filled with water at time t = 0.

For t > 0, volume v of solution containing mass m of solute flows into tank A per second; mixture flows from tank A to tank B at the same rate; and mixture flows away from tank B at the same rate. The differential equations used to model this system are given by dσA dt +

v V

σA =

m V

, dσB dt +

v V

σB =

v V

σA, where σA,B are the concentrations of solute in tanks A and B, respectively.

Show that the mass of solute in tank B is given by mV v

1 − e−vt/V

− mte−vt/V .