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5.5: Exercises- Matrix Methods for Dynamical Systems

Last updated

Sep 17, 2022
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- 5.4: The Backward-Euler Method
- 5.6: Supplemental - Matrix Analysis of the Branched Dendrite Nerve Fiber

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Exercise $\PageIndex{1}$

Compute, without the aid of a machine, the Laplace transforms of $e^t$ and $te^{-t}$ . Show ALL of your work.

Exercise $\PageIndex{2}$

Extract from fib3.m analytical expressions for $x_2$ and $x_{3}$

Exercise $\PageIndex{3}$

Use eig to compute the eigenvalues of $B = \begin{pmatrix} {2}&{-1}\\ {-1}&{2} \end{pmatrix}$ . Use det to compute the characteristic polynomial of $B$ roots to compute the roots of this characteristic polynomial. Compare these to the results of eig. How does Matlab compute the roots of a polynomial? (type help roots for the answer).

Exercise $\PageIndex{4}$

Adapt the Backward Euler portion of fib3.m so that one may specify an arbitrary number of compartments, as in fib1.m. Submit your well documented M-file along with a plot of $x_{1}$ and $x_{10}$ versus time (on the same well labeled graph) for a nine compartment fiber of length $l = 1cm$ .

Exercise $\PageIndex{5}$

Derive $\frac{\tilde{x}(t)-\tilde{x}(t-dt)}{dt} = B \tilde{x}(t)+g(t)$ from $\textbf{x}' = B \textbf{x}+\textbf{g}$ , by working backwards toward $x(⁢0)$ . Along the way you should explain why

$\frac{(\frac{I}{d(t)}-B)^{-1}}{d(t)} = (I-d(t)B)^{-1}$

Exercise 5.5.1\PageIndex{1}

Exercise 5.5.2\PageIndex{2}

Exercise 5.5.3\PageIndex{3}

Exercise 5.5.4\PageIndex{4}

Exercise 5.5.5\PageIndex{5}

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Exercise $\PageIndex{1}$

Exercise $\PageIndex{2}$

Exercise $\PageIndex{3}$

Exercise $\PageIndex{4}$

Exercise $\PageIndex{5}$