Tutorial Week 5
From UBCMATH WIKI
Worksheet Questions, February 6, 2017
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- 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).
- 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.
- 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.
- Consider the matrix $A = \left(\begin{array}{cc}
3\alpha - 8 & 7 \\
-4 & 3\alpha + 8
\end{array}\right).$
- Find the eigenvalues and eigenvectors of $A$.
- 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?
- Sketch the eigenvectors and some solution curves of the system $\mathbf{x'} = A\mathbf{x}$ in the case when $\alpha = 0$.
