9.\\
\includegraphics[max width=\textwidth, alt={}, center]{ab2949b2-11f2-4682-ab0c-25ecee2d665a-4_268_648_1169_623}

Two tanks, $A$ and $B$, each have a capacity of 800 litres.

At time $t = 0$ both tanks are full of pure water.

When $t > 0$, water flows in the following ways:

\begin{itemize}
  \item Water with a salt concentration of $\mu$ grams per litre flows into tank $A$ at a constant rate
  \item Water flows from tank $A$ to tank $B$ at a rate of 16 litres per minute
  \item Water flows from tank $B$ to tank $A$ at a rate of $r$ litres per minute
  \item Water flows out of tank $B$ through a waste pipe
  \item The amount of water in each tank remains at 800 litres.
\end{itemize}

This system is represented by the coupled differential equations

$$\begin{aligned}
& \frac { \mathrm { d } x } { \mathrm {~d} t } = 36 - 0.02 x + 0.005 y \\
& \frac { \mathrm {~d} y } { \mathrm {~d} t } = 0.02 x - 0.02 y
\end{aligned}$$

Solve the coupled differential equations to find both $x$ and $y$ in terms of $t$.\\

\hfill \mbox{\textit{SPS SPS FM 2020 Q9 [9]}}