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A Simple Harmonic Motion Question and Particular Solutions

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After working through the question in the pictures I have two questions.

Firstly how did they plot the equation and know which parts were amplitude ect?

Secondly how did they find the particular solution? I am not really sure where they got the substitution from in that question.

Thank you!

Re: A Simple Harmonic Motion Question and Particular Solutions

The second question first: Method of undetermined coefficients.

For the first, have you ever encountered the term "envelope?" (Sorry, couldn't find a good definition from the web.) An envelope is kind of a limiting solution. In this case it's a sine wave. As your solution mentions, the sin(delta*t) term has a very long wavelength compared to the other sine term. So sketch the sin(delta*t) term. Then we have a varying amplitude:

(or so since delta is so small.) This will be a "wave inside a wave." How can you tell? Either graph it using, say, Wolfram|Alpha or you have seen this kind of problem before.

-Dan