USE MATLAP:

In real life we are never given a differential equation to solve. We must find the differential equation based

on what we know about the process and physical laws. Let's look at the tank problem that we did in Week

$8,$ but this time with a slight twist.

Two tanks with capacities of $10$ liters initially contain $2$ grams of salt and $1$ liter of water. Water containing

$1\frac{g}{L}$ of salt enters the first tank at a rate of $2\frac{L}{?}$ hour, and the well$-$mixed solution flows out of the first tank

into the second tank at a rate equivalent to the volume of the first $($i$.$e$.,V_1\frac{L}{H}r).$ None of the brine flows

out of the second tank.

To model the process follow the steps in the lecture for each tank separately. Notice that the rate for the

second tank will depend on the first.

$(1)$ Model the problem for the volume in liters as an IVP$($s$)$ and by using ode$45($for MATLAB$)$ or

solve$\_$ivp $($for Python$)$ solve the IVP $($ODE $+$ IC$)$ for the volume of water in the tanks. Solve from

time $t=0$ to $t=6$ with $t=0\mathrm{.}01.$ Save the volume of the first tank at each time as a $6011$ column

vector named A$1$ and the volume of the second tank at each time as a $6011$ column vector named

A$2.$

$(2)$ At what time to the nearest integer does the water start overflowing $($hint: use the round function$)?$

Save this integer value as A$3.$ And which tank does it overflow from $(1$ or $2)?$ Save this integer value

as A$4.$

$(3)$ Model the problem for the amount of salt in grams as an IVP$($s$)$ and by using ode$45($for MATLAB$)$

or solve$\_$ivp $($for Python$)$ solve the IVP $(ODE+IC)$ for the amount of salt in the tanks. Solve from

time $t=0$ to $t=6$ with $t=0\mathrm{.}01.$ Save the amount of salt of the first tank at each time as a $6011$

column vector named A$5$ and the amount of salt of the second tank at each time as a $6011$ column

vector named A$6.$

$(4)$ What is the salt concentration when tank that overflowed in part $($b$)$ is completely full of brine? Save

this salt concentration $($remember it's amount$/$volume$)$ as A$7.$