Differential equations can also be used to model mixing or
dilution problems. If we call $y$ the quantity of a given
substance in a solution, then its rate of change with respect to
time $t$ will be given by $$ \frac{dy}{dt}=\text{ flow in }-\text{
flow out}. $$ On the other hand, $$ \text{ flow }= \text{
concentration } \times \text{ velocity}, $$ and $$ \text{
concentration } = \frac{ \text{ quantity } }{\text{ volume }}. $$
(Notice that the volume might depend on $t$ as well, such as in
case the velocities of flow in and flow out are different.) This
gives us

$$ \frac{dy}{dt}
=\frac{y \text{ coming in }}{\text{ volume }}\times \text{
velocity in } -\frac{y \text{ going out }}{\text{ volume
}}\times \text{ velocity out}. $$

Example: Lake
Pristine holds 50,000 m$^3$ of water. An accident at a
nearby factory dumps 10 tons of toxic waste into the lake,
where it dissolves. Every day, 5,000 m$^3$ of clean water
flows into the lake and 5,000 m$^3$ of lake water flows out.
How much waste will remain in the lake in 30 days?