What is the differential equation for the problem below?

You want an differential equation that describes the net rate of change in the amount of water in the pool.  In words,:

dA(t)/dt = net rate of change = rate water is coming in - rate water is going out.

where A(t) is the amount of water in the pool at time t.

Water is coming into the pool at a constant rate of 150 gal/day, so the first term on the left hand side of the equation is simply 150 gal/day.

Water is evaporating at a rate proportional to the amount of water present at a given time, so

rate water is going out = 0.003*A(t)/day

Putting this together, we get the differential equation:

dA(t)/dt = 150 gal/day - 0.003*A(t)/day

This is the differential equation asked for in the question.  This is a separable equation:

dA/dt = -(0.003/day)*(A - 50,000 gal)

dA/(A - 50,000 gal) = -0.003/day

Integrate both sides:

ln(A - 50,000gal) - ln(Ao - 50,000 gal) = -0.003*t/day

where Ao is the amount of water in the pool at time t = 0.

ln((A - 50,000gal)/(Ao - 50,000 gal)) = -0.003*t/day


(A - 50,000gal)/(Ao - 50,000 gal) = exp(-0.003*t/day)

A(t) = 50,000gal - (Ao - 50,000 gal)*exp(-0.003*t/day)

This is the general solution to the differential equation.  Note that as time -> infinity, the exponential term goes to zero, and the volume of water in the pool approaches the steady-state value of 50,000 gal.  This is the volume at which the evaporation rate exactly balances the rate that water is being added to the pool (i.e., the value of A for which dA/dt = 0).