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5.5: Exercises- Matrix Methods for Dynamical Systems

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    21830

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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

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

    This page titled 5.5: Exercises- Matrix Methods for Dynamical Systems is shared under a CC BY 1.0 license and was authored, remixed, and/or curated by Steve Cox via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.