Basic Differential Equations Suppose there are two lakes located on a stream. Clean water flows into the first lake, then the water from the first lake flows into the second lake, and then water from the second lake flows further downstream. The in and out flow from each lake is 500 liters per hour. The first lake contains 100 thousand liters of water and the second lake contains 200 thousand liters of water. A truck with 500 kg of toxic substance crashes into the first lake. Assume that the water is being continually mixed perfectly by the stream. a) Find the concentration of toxic substance as a function of time in both lakes. b) When will the concentration in the first lake be below 0.001 kg per liter? c) When will the concentration in the second lake be maximal? I'm completely stuck on how to do the question.

I'll help with the first lake.

For the sake of understanding, let's call the respective volume of lakes $Q_1,\,Q_2$, the flows are $v$, the mass of poison in lakes $m_i$. Initial mass of poison $m$.

The concentration in the first lake is $\frac{m_1}{Q_1}$, hence the flow brings in $v\Delta t$ of fresh water and carries away $v\Delta t \frac{m_1}{Q_1}$ of toxic substance. Thus, $\Delta m_1 = v\Delta t \frac{m_1}{Q_1}$, or, in other words, $$\frac{d}{dt}m_1(t) = m_1(t)\frac{v}{Q_1},\quad m_1(0)=m.$$ We can solve this equation.

Can you take it from here and apply the same method to obtain the equation for the second lake?