Consider the two tanks shown in the figure below. Assume that tank A contains 50 gallons of water in which 25 pounds of salt is dissolved. Suppose tank B contains 50 gallons of pure water. Liquid is pumped into and out of the tanks as indicated in the figure; the mixture exchanged between the two tanks and the liquid pumped out of tank B are assumed to be well stirred. We wish to construct a mathematical model that describes the number of pounds x1(t) and x2(t) of salt in tanks A and B, respectively, at time t. This system is described by the system of equations
dx1/dt = − 2/25(x1) + 1/50(x2)
dx2/dt= 2/25(x1) − 2/25(x2)
with initial conditions
x1(0) = 25,
x2(0) = 0
(see (3) and the surrounding discussion on mixtures on page 107). What is the system of differential equations if, instead of pure water, a brine solution containing 4 pounds of salt per gallon is pumped into tank A?