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Oscillator response function

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Obtain the response of a linear oscillator to a step function and to an impulse function (in the limit t goes to infinity) for over-damped oscillator. Sketch the response functions.

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Solution Summary

The 9 pages solution includes full derivations of all equations and also includes a derivation of the variation of parameters method to solve a second order ODE.

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Here is the solution.
The first two pages are simple reminder of the variation of parameters method to solve differential equations.
I use it to solve the equation of motion of the oscillator.

I added some other analysis as to where the impulse response gets it maxima and what it the step function asymptotic response.

Let's first develop the general course of action when dealing with variation of parameters.
Assume the equation is in the form:

And also assume we know the homogenous solutions .
We guess a particular solution in the form
Therefore

Also, since are arbitrary functions, we can ...

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