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Most DE mixing problems ask you to solve for y(t) out - that is, concentration or rate out. I have no problems with these but cannot see how to set up mixing problem where you need to solve for unknown concentration in. For example: Tank 200 l fresh water. Rate of salt in and out = 2 L/min. Concentration salt in tank at 60 min = 1.4 gm/L. Find concentration of salt entering tank. I only need to know how to set up the DE.

The tank is 200 L fresh water.

Salt water is added to the tank 2 L/min.

Water already in the tank is drained out at 2 L/min.

When the time t is 60 min, the concentration of salt 1.4 gm/L

The incoming rate gives us dy/dt = 2a/200 = a/100 where a needs to be found.

The outgoing rate gives us dy/dt = 0.01y.

It is given that originally, we have no concentration, so y(0) = 0.

It then says that in 60 seconds the concentration is 1.4, so y(60) = 1.4

The equation is y(t) = A-Be^(-Ct).

Since y(0) = 0 and from above, y(0) = A-B, that says 0 = A - B, so A = B.

That gives y(t) = A(1-e^(-Ct)).

Since the rate is 0.01, and the rate is also C, that says C = 0.01.

That gives y(t) = A(1-e^(0.01t)).

It says that at t=60, y=1.4, so we have y(60) = 1.4.

This says that 1.4 = A(1-e^0.6), which means A = 1.4/(1-e^0.6).

Substituting this back into the equation for y gives

y(t) = 1.4((1-e^-0.01t)/(1-e^0.6).

Letting t go go oo, it can be seen that y(t) goes to a 1.4/(1-e^0.6), which is A.

Note that A is slightly over 3.102917.

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Comment | Thanks so much. That was even more than I expected and I was well able to understand how to set up the problem. |

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