Chapter 9.5, Problem 33E

In Section 9.3 we looked at mixing problems in which the volume of fluid remained constant and saw that such problems give rise to separable differentiable equations. (See Example 6 in that section.) If the rates of flow into and out of the system are different, then the volume is not constant and the resulting differential equation is linear but not separable.

A tank contains 100 L of water. A solution with a salt concentration of 0.4 kg/L is added at a rate of 5 L/min. The solution is kept mixed and is drained from the tank at a rate of 3 L/min. If $y\left(t\right)$ is the amount of salt (in kilograms) after t minutes, show that y satisfies the differential equation

$\frac{dy}{dt}=2-\frac{3y}{100+2r}$

Solve this equation and find the concentration after 20 minutes.