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**Prove It** Your calculation of $\displaystyle \frac{dy}{dx}$ is correct.

So $\displaystyle \frac{d^2y}{dx^2} = \frac{d}{dx}\left(e^{-t}\,\frac{dy}{dt}\right)$

$\displaystyle = e^{-t}\,\frac{d}{dx}\left(\frac{dy}{dt}\right) + \frac{d}{dx}(e^{-t})\,\frac{dy}{dt}$

$\displaystyle = e^{-t}\,\frac{d}{dt}\left(\frac{dy}{dt}\right)\,\frac{ dt}{dx} + \frac{d}{dt}(e^{-t})\,\frac{dt}{dx}\,\frac{dy}{dt}$

$\displaystyle = e^{-2t}\,\frac{d^2y}{dt^2} - e^{-2t}\,\frac{dy}{dt}$.

Now substitute everything into your DE.