## Mathematics & Science
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## Applications of Differential Equations## The Simple Pendulum
Your output from the command I just gave you should have looked something like this:
Series[Sin[y],{y,0,10}] The last term shown here is called the order term, and is read as "big oh of
Now, look at the size of the denominators of each term the further you go out in the series. Here's some initial evidence that the terms are shrinking in size as you move out into the series' higher degree terms. But hold on! This affect could be spoiled by the exponents in the series. After all, big powers usually mean big numbers, and these exponents are growing in size. How do we know that they're not growing so quickly as to defeat the shrinking affect produced by the denominators. Well, one way we can guarantee some size control is by restricting the size
of
This is exactly the same conclusion that we reached when we took the
graphical approach. It seems that there is adequate evidence at this point to
allow ourselves to make the substitution of ## Making the SubstitutionOnce we make the replacement mentioned above, our initial value problem goes from:
## Solving the Linearized ProblemNow we have a The problem would be a little easier to think about if we were to use
concrete values for our two constants, so let's set
the values of
Before we have y0=0, and v0=1 Note the use of single, not double, equals signs here since we're making an
assignment, not declaring an equation. Go to Let's now discuss your result... |
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ODE Laboratories: A Sabbatical Project by Christopher A. Barker©2017 San Joaquin Delta College, 5151 Pacific Ave., Stockton, CA 95207, USA e-mail: cbarker@deltacollege.edu |