How to write down the initial value problem for the filtering process?

Your swimming pool containing 60,000 gal of water has been contaminated by 5 kg of a nontoxic dye that leaves a swimmer's skin an unattractive green. The pool's filtering system can take water from the pool, remove the dye, and return the water to the pool at a flow rate of 200gal/min. Write down the initial value problem for the filtering process; let q(t) be the amount of dye in the pool at any time t.How to write down the initial value problem for the filtering process?
The differential equation is conceptually





dq/dt = (rate of q in) - (rate of q out)





The rate of q in is zero, because the filtering removes all the dye.





The rate of q out is equal to the concentration (kg/gal) times the flow rate (gal/min). At time t there are q kg of dye in 60000 gallons of pool water, so the concentration is q/60000 kg/gal. The flow rate is given as 200 gal/min, so the rate of q out is (q/60000)(200) = q/300 kg/min.





So the ODE is





dq/dt = -q/300





with initial condition q(0) = 5