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Math Help - Differential Equation Mixing Problem

  1. #1
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    Differential Equation Mixing Problem

    I'm having a bit of trouble with some questions, I'm not sure whether it is from deficiancies in D.E or integration and would like some help if possible.

    so here's an example.

    A tank with 200L of pure water initially, has 4grams of salt per litre run into the tank at 1L per min. A well mixed solution is drained out at 3L per min.

    I am to work out the amount of salt in the tank after t mins. Here's what im doing:

    V=200-2t--------volume function of time

    Dx/Dt=4-(3x/200-2t)------transpose

    Dx/dt+(3x/200-2t)=4

    integrating factor= e^((integral of)3/200-2t)
    =(200-2t)^(-3/2)

    Equation then becomes

    x(200-2t)^(-3/2)={integral of}4((200-2t)^(-3/2))
    Transposing
    x =4(200-2t)^(-.5)+c

    This equation doesn't seem to be realistic as shouldn't the amount of salt in the container look like a parabola or trig(sin/cos) function?

    Thanks in advance
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  2. #2
    Junior Member
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    You are absolutely right

    dy/dt = rate coming in - rate coming out
    dy/dt = 1*0.004-3y/200
    dy/dt =(-0.015)(y-4/15)
    dy/(y-4/15) = (-0.015)dt
    take integral of both side
    ln|y-4/15| = -0.015t
    y = e^{-0.015t}+4/15
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  3. #3
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    Quote Originally Posted by rubic View Post
    You are absolutely right

    dy/dt = rate coming in - rate coming out
    dy/dt" alt="dy/dt" /> = 1*0.004-3y/200
    dy/dt =(-0.015)(y-4/15)
    dy/(y-4/15) = (-0.015)dt
    take integral of both side
    ln|y-4/15| = -0.015t
    y = e^{-0.015t}+4/15

    Rate out is=ConcentrationxVolume

    C=y/(200-2t) v=3

    so shouldn't rate out =3y/(200-2t) as the volume is changing with respect to time?
    I should also mention that x, or y as you used is in grams.
    thanks
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  4. #4
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    so shouldn't rate out =3y/(200-2t) as the volume is changing with respect to time?

    oh yes, you are right..
    I try to solve again, and i had soln as a straight line
    thats what i found
    y = 0.8 - 0.008t

    i am no longer sure about it..
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