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Math Help - Another ODE

  1. #1
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    Another ODE

    I've attached the relevant pictures. The question is:
    Let X,Y be two containers.
    At t=0, container X has 100 lt. of water with 2 kg of salt in it and Y has 100 lt. of water with 6 kg of salt.
    On each t>0, the system transports water as the you can see in the picture.

    In each minute t, let x(t), y(t) be the quantities of salt in X,Y in kg's.
    t is measured in minutes!

    You should notice that on each time, there are excatly 100 lt. in each container!

    Write an ODE that gives the quantity of salt on each container as a function of time, solve it and calculate how much kg's of salt will be in the container after 10 minutes from the start of the process.
    My attemt:

    I wrote the equations this way:
    x'(t)= -8x(t)/100 +2y(t)/100
    y'(t) = 8x(t)/100 -8y(t)/100

    We get the ODE: w' =Aw ... The eignvalues of A are: -4/100 and -12/100 ...
    After I solve these two equations I get two soloutions- one for x(t) and one for y(t)...The only problem is that these soloutions don't match the data of the question...
    HELP is needed!

    TNX
    Attached Thumbnails Attached Thumbnails Another ODE-ode1.jpg  
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  2. #2
    MHF Contributor Calculus26's Avatar
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    Those are the correct eigenvalues

    Eigen vectors are for -1/25 (-1/2,1)

    -3/25 (1/2,1)

    applying the ICs

    x= 5/2*e^(-1/25t) -1/2 e^(-25/t)

    y= e^(-1/25t) + e^(-3/25t)

    See attachment
    Attached Thumbnails Attached Thumbnails Another ODE-eigenvals.jpg  
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  3. #3
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    Wow, so I was right... TNX a lot for your verification!
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