Tutorial Week 5

Worksheet Questions, February 6, 2017

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1. Salt water with concentration of $K$ in g/L flows into Tank A at a rate $a$ in L/min. The mixed solution in Tank A flows into Tank B at a rate $b$ L/min. Another pipe takes the solution in Tank B back into Tank A at a rate $c$ in L/min. Finally, solution drains out of Tank B at a rate $a$ in L/min. Note: the initial volume $V$ in each tank is the same and $a + c = b$ (to ensure that the volumes in the tanks are constant).
   1. Write down the system in matrix form that gives the amount of salt in each tank (call these $A(t)$ and $B(t)$) at any given time.
   2. Show that the eigenvalues of the matrix for this system are real, and therefore it is impossible for this system to have oscillations (damped or otherwise). \textit{Hint}: express the matrix in terms of $b$ and $c$ only. Optional: show that both eigenvalues are negative.

1. Consider the matrix $A = \left(\begin{array}{cc} 3\alpha - 8 & 7 \\ -4 & 3\alpha + 8 \end{array}\right).$
   1. Find the eigenvalues and eigenvectors of $A$.
   2. For what values of $\alpha$ (i) are both eigenvalues postive, (ii) are both eigenvalues negative, (iii) do the eigenvalues have opposite sign, (iv) is one eigenvalue zero?
   3. Sketch the eigenvectors and some solution curves of the system $\mathbf{x'} = A\mathbf{x}$ in the case when $\alpha = 0$.

Solutions:

1. 8pts total
   1. 2 pts for form (constant coefficient linear system with inhomogeneous terms), 2 pts for the details of the terms (A, B) so total 6 pts. \begin{equation} \begin{pmatrix} A'(t) \\ B'(t) \end{pmatrix}= \begin{pmatrix} -b/V & c/V \\ b/V & -(a + c)/V \end{pmatrix} \begin{pmatrix} A(t) \\ B(t) \end{pmatrix} + \begin{pmatrix} K a \\ 0 \end{pmatrix} \end{equation}
   2. 2pts By using $b = a + c$, eigenvalues are (1pt) $\frac{-b \pm \sqrt{bc}}{V}.$ Since $b \ge 0$ and $c \ge 0$ these are real (1pt - must be explained).
2. 14pts total
   1. 6pts Eigenvalues are $3\alpha \pm 6$ (2pts) with corresponding eigenvectors (4pts) $e_1 = \begin{pmatrix}1\\2\end{pmatrix}$ and $e_2 = \begin{pmatrix}7\\2\end{pmatrix}.$
   2. 4pts - one for each category There is a zero eigenvalue for $\alpha = \pm 2$, the eigenvalues are both negative for $\alpha < -2$, the eigenvalues are both positive for $\alpha > 2$, the eigenvalues have mixed signs for $-2 < \alpha < 2$.
   3. 4pts Sketch of eigenvectors from (a) worth 2pts and sketch of solution curves as below 2pts.