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# 5.5: Exercises- Matrix Methods for Dynamical Systems

• • Contributed by Steve Cox
• Emeritus Professor (Computational and Applied Mathematics) at Rice University
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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

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