The equation to be solved still is given by: $$m\,\ddot{x} + D\,x = 0$$ what can be easily resolved for $x$ using characteristic polynomials or other techniques customized for homogenous differential equations. I just ask myself whether there exist also an elegant way using Fourier Transform. Some of these steps have been unrolled in an equal question regarding this method but some of them are still unclear to me:
First of applying FT: $x(t) \to \hat{x}(\omega)$: $$-m\omega^2\,\hat{x}(\omega)+ D\,\hat{x}(\omega) = 0$$
Now usually this can be changed into an equation for $\hat{x}(\omega)$ solely on one side that can be inverse Fourier transformed but in this case I only get $\hat{x}(\omega) = 0$ what is kind of leading to nowhere. Any escape?