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Help with num 12 please. thanks​

Help with num 12 please. thanks​-example-1
User RaR
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1 Answer

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Explanation:

Given:


x = e^(-t)\sin t

Taking the 1st and 2nd derivatives of the above expression,


(dx)/(dt) = -e^(-t)\sin t + e^(-t)\cos t


(d^2x)/(dt^2) = e^(-t)\sin t - e^(-t)\cos t -e^(-t)\cos t


\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:- e^(-t)\sin t


\:\:\:\:\:\:\:\:\:= -2e^(-t)\cos t

Therefore,


(d^2x)/(dt^2) + 2(dx)/(dt) + 2x


= -2e^(-t)\cos t + 2(-e^(-t)\sin t + e^(-t)\cos t)


\:\:\:\:+ 2e^(-t)\sin t


= -2e^(-t)\cos t - 2e^(-t)\sin t + 2e^(-t)\cos t + 2e^(-t)\sin t


= 0

This shows that
x = e^(-t)\sin t is the solution to the differential equation


(d^2x)/(dt^2) + 2(dx)/(dt) + 2x = 0

User Adrian Krebs
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