Let y(t) denote the amount of salt in the tank at time t. Its time rate of change is

(1)

5 lb times 10 gal gives an inflow of 50 lb of salt. Now, the outflow is 10 gal of brine. This is 10/1000=0.01 (=1%) of the total brine content in the tank, hence 0.01 of the salt content y(t), that is, 0.01y(t). Thus, from (1) we obtain the following ODE as a model:

.

(2)

The ODE (2) is separable. Separation, integration, and taking exponents on both sides gives

, , .

Initially, the tank contains 100 lb of salt. Hence y(0)=100 is the initial condition that will give the unique solution. Substituting y=100 and t=0 in the last equation gives 100-5000=ce⁰=c. Hence c=4900. Hence the amount of salt in the tank at time t is

.

(3)

This function (see the graph on the right) shows an exponential approach to the limit 5000 lb. Can you explain physically that y(t) should increase with time? That its limit is 5000 lb. Can you see the limit directly from the ODE?