5.5: Exercises- Matrix Methods for Dynamical Systems
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Compute, without the aid of a machine, the Laplace transforms of et and te−t. Show ALL of your work.
Extract from fib3.m
analytical expressions for x2 and x3
Use eig
to compute the eigenvalues of B=(2−1−12). 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).
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 x1 and x10 versus time (on the same well labeled graph) for a nine compartment fiber of length l=1cm.
Derive ˜x(t)−˜x(t−dt)dt=B˜x(t)+g(t) from x′=Bx+g, by working backwards toward x(0). Along the way you should explain why
(Id(t)−B)−1d(t)=(I−d(t)B)−1