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Andrei [34K]
2 years ago
11

Help with num 12 please. thanks​

Mathematics
1 answer:
RoseWind [281]2 years ago
8 0

Step-by-step explanation:

Given:

x = e^{-t}\sin t

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

\dfrac{dx}{dt} = -e^{-t}\sin t + e^{-t}\cos t

\dfrac{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,

\dfrac{d^2x}{dt^2} + 2\dfrac{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

\dfrac{d^2x}{dt^2} + 2\dfrac{dx}{dt} + 2x = 0

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