Consider two interconnected tanks as shown in the figure above. Tank 1 initial contains 30 L (liters) of water and 175 g of salt, while tank 2 initially contains 60 L of water and 415 g of salt. Water containing 50 g/L of salt is poured into tank1 at a rate of 1.5 L/min while the mixture flowing into tank 2 contains a salt concentration of 30 g/L of salt and is flowing at the rate of 3.5 L/min. The two connecting tubes have a flow rate of 3.5 L/min from tank 1 to tank 2; and of 2 L/min from tank 2 back to tank 1. Tank 2 is drained at the rate of 5 L/min.

You may assume that the solutions in each tank are thoroughly mixed so that the concentration of the mixture leaving any tank along any of the tubes has the same concentration of salt as the tank as a whole. (This is not completely realistic, but as in real physics, we are going to work with the approximate, rather than exact description. The 'real' equations of physics are often too complicated to even write down precisely, much less solve.)

How does the water in each tank change over time?

Let $p(t)$ and $q(t)$ be the number of grams (g) of salt at time t in tanks 1 and 2 respectively. Write differential equations for $p$ and $q$. (As usual, use the symbols $p$ and $q$ rather than $p(t)$ and $q(t)$.)
$p'=$
$q'=$

Give the initial values
$\begin{bmatrix} p(0) \\ q(0) \end{bmatrix}$ = $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$

Show the equation that needs to be solved to find a constant solution to the differential equation:
$\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$$=$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$ $\begin{bmatrix} p \\ q \end{bmatrix}$

A constant solution is obtained if $p(t) =$ for all time t and $q(t) =$ for all time t.