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# Differential Equations/DE mixing problem with unknown concentration in

Question
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.

Questioner's Rating
 Rating(1-10) Knowledgeability = 10 Clarity of Response = 10 Politeness = 10 Comment Thanks so much. That was even more than I expected and I was well able to understand how to set up the problem.

Differential Equations

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#### Scott A Wilson

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I am capable of solving most ordinary differential equations and a few not so ordinary, but those are few and far between.

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I have taken Differential Equations at OSU. Occasionally I have assisted people with questions on the subject.

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I wrote a thesis on the study of shock waves and rarefaction fans. That occurs in nature when mathematics fails to apply. See, mathematics says that when shock waves appear, there should be two solutions whereas nature knows there is only one way to be in that area. When rarefaction fans occur, the mathematics says there should be no solution, but despite this, nature still moves ahead with its own plan.

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I received an MS degree at OSU in Mathemematics and a BS degree at OSU in Mathematical Science.

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I graduated from OSU with a Bachelor of Science degree in mathematics with honors. Two years later, I graduated from OSU with a Master of Science degree in mathematics with honors. Between these two degrees, I have more hours in graduate courses that required to get them.

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I answered many questions at OSU, down in south seattle at a college, at a church in Corvallis, OR, as Safeway in Washington, and many other areas. I have answered over 8,500 right here, but only a little over 190 of them have been in differential equations.