 ## Applications of Differential Equations

### The Simple Pendulum

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Your output from the command I just gave you should have looked something like this:

Plot[{Sin[y],y},{y,-0.5,0.5}] What do you notice about the two graphs? Hopefully you notice that they are barely distinguishable from one another. Now certainly the graphs of sin(y) and y are not this similar for all values of y. The values that we used, -0.5≤y≤0.5, are in the neighborhood of zero. The picture that we just made seems to support the following claim:

for small values of y, sin(y) is approximately equal to y

Now as mathematicians we're always a little suspicious of how much trust we can put in a picture, so now let's look at another approach that leads us to the same conclusion.

The Series Approach: Do you remember the idea of expanding a function into an infinite series using Taylor's Theorem? You probably learned about it in your second calculus course.

We're going to expand sin(y) into a series about the origin. Thankfully Mathematica can do this work for us. The command that we'll use is this:

Series[Sin[y],{y,0,10}]

Here the zero tells Mathematica where to center the series, and the 10 tells it up to what degree terms it should continue the series expansion.

Now go to Mathematica and enter this new command.

Let's now discuss your result... If you're lost, impatient, want an overview of this laboratory assignment, or maybe even all three, you can click on the compass button on the left to go to the table of contents for this laboratory assignment.

ODE Laboratories: A Sabbatical Project by Christopher A. Barker

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