# That the second order differential equation which describes the motion

August 31, 2017

Question
GIVEN: That the second order differential equation which describes the motion of the pendulum is nonlinear, so at that time, we linearized aboutϴ= 0 which means that the solution will be good only for small angles. The differential equation is;

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Let;

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Therefore, the differential equation is;

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1) Show the steps in using the change of variables

.png”>

to convert the above nonlinear, second order differential equation (the one with

numbers) to the first order system;

.png”>

2) Numerically solving the differential equation with the initial conditions;

.png”>

and then creating graph of, below;

.png”>

.png”>

Use this graph to discuss the actual motion of the pendulum in the time interval .png”>

That is, where is the pendulum at time;

.png”>

Then as time increases, is the pendulum moving clockwise or counterclockwise and for approximately how long? ETC.

3) Numerically solving the differential equation with the

given initial conditions and then creating the graphs of;

.png”>

in each case (keep in mind that;

.png”>

Using the graphs to think about the actual motion of the pendulum, explain why the

graphs appear as they do. In particular, you should be able to explain why;

.png”>

in the second graph.

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