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, $\displaystyle x_0$kg of a chemical X is dissolved into tank A, and tank B has $\displaystyle y_0$kg of the same chemical X dissolved into it.I put this: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.

$\displaystyle x(t)=\frac{-5x_0}{200}+\frac{5y_0}{200}=\frac{1}{40}(-x_0+y_0)$

$\displaystyle y(t)=\frac{5x_0}{200}-\frac{5y_0}{200}=\frac{1}{40}(x_0-y_0)$

$\displaystyle \frac{1}{40}\begin{pmatrix}ii). Find the eigenvalues and the eigenvectors of the resulting matrix form.

{-1}&{1}\\

{1}&{-1}

\end{pmatrix}\begin{pmatrix}

{x_0}\\

{y_0}

\end{pmatrix}=\begin{pmatrix}

{x(t)}\\

{y(t)}

\end{pmatrix}$

$\displaystyle \frac{1}{40}\begin{vmatrix}

{-1-\lambda}&{1}\\

{1}&{-1-\lambda}

\end{vmatrix}=\frac{1}{40}((1+\lambda)^2-1)=0$

$\displaystyle (1+\lambda^2)-1=0$

$\displaystyle \lambda^2+2\lambda=0 \Rightarrow \lambda(\lambda+2)=0$

The eigenvalues are 0 and -2.

$\displaystyle \lambda=0$, Eigenvector:$\displaystyle \begin{pmatrix}

{-1}\\

{1}

\end{pmatrix}$

$\displaystyle \lambda=-2$, Eigenvector: $\displaystyle \begin{pmatrix}

{1}\\

{1}

\end{pmatrix}$

$\displaystyle X(t)=A\begin{pmatrix}iii). Show that the amount of the chemical X in either tank approaches $\displaystyle \frac{1}{2}(x_0+y_0)$ as t approaches infinity.

{-1}\\

{1}

\end{pmatrix}+Be^{-2t}\begin{pmatrix}

{1}\\

{1}

\end{pmatrix}$

At t=0:

$\displaystyle x(0)=x_0$

$\displaystyle y(0)=y_0$

$\displaystyle x_0=B-A$

$\displaystyle y_0=A+B$

Working this out gives:

$\displaystyle B=\frac{1}{2}(x_0+y_0)$

$\displaystyle A=\frac{1}{2}(y_0-x_0)$

$\displaystyle X(t)=\frac{1}{2}(y_0-x_0)\begin{pmatrix}

{-1}\\

{1}

\end{pmatrix}+\frac{1}{2}(x_0+y_0)e^{-2t}\begin{pmatrix}

{1}\\

{1}

\end{pmatrix}$

as $\displaystyle t \rightarrow \infty$:

$\displaystyle X(t) \rightarrow \begin{pmatrix}

{\frac{1}{2}(x_0+y_0)}\\

{\frac{1}{2}(y_0-x_0}

\end{pmatrix}$

I get really far but this doesn't work out. It suggests that my inital formulae are wrong but I can't see where my mistake is. If I reversed my A and B my formulae would work, but I can't see how I can do this.

Help would be appreciated greatly!