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Math Help - Dissolve question

  1. #1
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    Dissolve question

    Consider two tanks, A and B, each holding 200 litres of water. A pipe pumps water from tank A to tank B at a rate of 5l/min. At the same time another pipe pumps liquid from tank B to tank A at the same rate. At time t=0, kg of a chemical X is dissolved into tank A, and tank B has kg of the same chemical X dissolved into it.
    i). Write down the system of differential equations satisfied by x(t) and y(t), the quantity of the chemical X in tanks A and B respectively.
    x(t)=x0-t(y0/40-x0/40)
    y(t)=y0-(y0/40+x0/40)
    dx/dt=t(-y0/40+x0/40)
    dy/dt=t(y0/40-x0/40)
    ii). Find the eigenvalues and the eigenvectors of the resulting matrix form.
    The matrix is
    (-1/40 1/40)
    (1/40 -1/40)
    The eigenvalues are 0 and -2
    so the eigenvector are
    (-1)
    (1 )
    and
    (1)
    (1)

    iii). Show that the amount of the chemical X in either tank approaches as t approaches infinity.
    i dont have any idea to do this part, can anyone help me? and tell me what i did wrong in the first and second part pls
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  2. #2
    A Plied Mathematician
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    CT, USA
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    Re: Dissolve question

    Quote Originally Posted by kanezila View Post
    Consider two tanks, A and B, each holding 200 litres of water. A pipe pumps water from tank A to tank B at a rate of 5l/min. At the same time another pipe pumps liquid from tank B to tank A at the same rate. At time t=0, kg
    Did you mean to put a quantity of chemical X there? I don't see a number.

    of a chemical X is dissolved into tank A, and tank B has kg
    Same issue here.

    of the same chemical X dissolved into it.
    i). Write down the system of differential equations satisfied by x(t) and y(t), the quantity of the chemical X in tanks A and B respectively.
    x(t)=x0-t(y0/40-x0/40)
    y(t)=y0-(y0/40+x0/40)
    dx/dt=t(-y0/40+x0/40)
    dy/dt=t(y0/40-x0/40)
    Why are there t's in these equations? What is your justification for your answer of this part of the problem?

    ii). Find the eigenvalues and the eigenvectors of the resulting matrix form.
    The matrix is
    (-1/40 1/40)
    (1/40 -1/40)
    I agree with your matrix form. That is, the DE can be written as

    \frac{d}{dt}\begin{bmatrix}x\\ y\end{bmatrix}=\begin{bmatrix}-1/40 &1/40\\ 1/40 &-1/40\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}.

    If you let

    A:=\begin{bmatrix}-1/40 &1/40\\ 1/40 &-1/40\end{bmatrix},

    then the solution to this differential equation is

    \begin{bmatrix}x\\ y\end{bmatrix}=e^{At}\begin{bmatrix}x_{0}\\ y_{0}\end{bmatrix}.

    What you must do is make sense of the exponential e^{At}. How do you do matrix exponentiation? This is how you can finish the problem.

    The eigenvalues are 0 and -2
    so the eigenvector are
    (-1)
    (1 )
    and
    (1)
    (1)

    iii). Show that the amount of the chemical X in either tank approaches as t approaches infinity.
    i dont have any idea to do this part, can anyone help me? and tell me what i did wrong in the first and second part pls
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