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Show that the equation x=asinomegat+bcos...

Show that the equation `x=asinomegat+bcosomegat` represents a simple harmonic motion.

Text Solution

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`x=asinomegat+bcosomegat`
or, `(dx)/(dt)=aomegacosomegat-bomegasinomegat`
or, `(d^(2)x)/(dt^(2))=-aomega^(2)sinomegat-bomega^(2)cosomegat`
or, `(d^(2)x)/(dt^(2))=-omega^(2)(asinomegat+bcosomegat)`
or, `(d^(2)x)/(dt^(2))=-omega^(2)xor,(d^(2)x)/(dt^(2))+omega^(2)x=0`
This is the differential equation of simple harmonic motion.
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