Question

Help with num 12 please. thanks​

Answer 1

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`