Question

Consider a flask that contain 3 liters of salt water. Suppose that water containing 25 grams per liters of salt is pumped into the flask at the rate of 2 liters per hour, and the mixture, being steadily stirred, is pumped out of the flask at the same rate. Find a differential equation satisfied by the amount of salt 𝑓(𝑡) in the flask at time 𝑡.

Answer 1

Let a = 25 grams/liter, V = 3 liters, u = 2 liters/hour

$df=a\cdot u\cdot dt-\frac{f}{V}u\cdot dt\\ \frac{df}{dt}=u(a-\frac{f}{V})\\ \int_0^f\frac{d(aV-f)}{aV-f}=-\int_0^t\frac{u}{V}dt\\ aV-f=(aV-f(0))e^{-\frac{u}{V}t}\\ f=aV(1-e^{-\frac{u}{V}t})=75(1-e^{-\frac{2}{3}t})~(g)$