Question

Two large​ tanks, each holding 100 L of​ liquid, are interconnected by​ pipes, with the liquid flowing from tank A into tank B at a rate of 3 ​L/min and from B into A at a rate of 1 ​L/min. The liquid inside each tank is kept well stirred. A brine solution with a concentration of 0.2 ​kg/L of salt flows into tank A at a rate of 6 ​L/min. The​ (diluted) solution flows out of the system from tank A at 4 ​L/min and from tank B at 2 ​L/min. If​ initially, tank A contains pure water and tank B contains 20 kg of​ salt, determine the mass of salt in each tank at time tgreater than or equals0.

Answer 1

Setting up the ODE system is perhaps the hardest part, so I'll do just that and include the general and particular solution at the end. (Showing all the work would definitely exceed the character limit.)

Let $A(t)$ and $B(t)$ denote the amount of salt in tanks A and B, respectively, at time $t$.

Salt flows into tank A at a rate of

(0.2 kg/L) * (6 L/min) + (B(t)/100 kg/L) * (1 L/min) = (1.2 + B(t)/100) kg/min

and flows out at a rate of

(A(t)/100 kg/L) * (4 + 3 L/min) = 7 A(t)/100 kg/min

Salt flows into tank B at a rate of

(A(t)/100 kg/L) * (3 L/min) = 3 A(t)/100 kg/min

and flows out at a rate of

(B(t)/100 kg/L) * (1 + 2 L/min) = 3 B(t)/100 kg/min

Then the net rate of change of the amount of salt in the tanks is governed by the system of ODEs,

$\begin{cases}A'=1.2-\frac{7A}{100}+\frac B{100}\\B'=\frac{3A}{100}-\frac{3B}{100}\end{cases}$

or in matrix form,

$\mathbf x'=\frac1{100}\begin{bmatrix}-7&1\\3&-3\end{bmatrix}+\begin{bmatrix}1.2\\0\end{bmatrix}$

From here I'd suggest solving using the method of undetermined coefficients.

See the screenshot for the particular solution to the system with the initial conditions $A(0)=0$ and $B(0)=20$.