Dampened Harmonic Oscillator via Laplace Transforms

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Question statement

Solve the differential equation

$x^{\prime\prime} = -\simplify{{k}*{k}}x - \var{b}x^\prime \quad \mbox{ with the initial conditions } \quad x(0) = \var{l} \, \mbox{ and }\, x^\prime(0) = 0$

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			Name	Type	Generated Value

k	integer	7
b	integer	14
l	integer	1

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"Part b)" → "Unnamed gap" - Correct answer
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"Part d)" → "Unnamed gap" - Correct answer

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Take the Laplace transform of the given ODE and write down the transformed equation:

Part a)

Advice

By re-arranging the given ODE we get:

$x^{\prime\prime} +\var{b}x^\prime + \simplify{{k}*{k}}x = 0.$

Then, let $X=\mathcal{L}\{x\}$ , and apply Laplace Transform to get

$(s^2X-sx(0)-x^\prime(0)) +\var{b}(sX-x(0)) +\simplify{{k}*{k}}X=0.$

Then plugging in the initial values:

$(s^2X-\var{l}s) +\var{b}(sX-\var{l}) +\simplify{{k}*{k}}X=0.$

By re-arranging, we get

$X=\frac{\var{l}s + \simplify{2*{k}{l}}}{s^2+\var{b}s +\simplify{{k}*{k}}} = \frac{\var{l}(s +\var{k}) + \simplify{{l}*{k}}}{(s+\var{k})^2} = \frac{\var{l}}{s+\var{k}} + \frac{\simplify{{l}*{k}}}{(s+\var{k})^2}.$

Finally, by the transform table and the property $\mathcal{L}\left\{t^n g(t)\right\} = (-1)^n\frac{d^nG}{ds^n}$ (where $\mathcal L\{g(t)\} = G(s)$ (Note that one can use the First Shift Theorem instead of this property here).

$x(t) =\var{l}e^{-4t} + \simplify{{l}*{k}}te^{-4t}$

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Laplace Transforms by Ugur Efem in Ugur's workspace.

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Dampened Harmonic Oscillator via Laplace Transforms

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Dampened Harmonic Oscillator via Laplace Transforms

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Ugur Efem 2 years, 1 month ago

Ugur Efem 2 years, 1 month ago

Ugur Efem 2 years, 1 month ago

Error in variable testing condition

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Generated value: integer

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Generated value: `integer`