Fourier transform of the density fluctuation

Question

There is a Fourier transform that I don't really understand in my textbook (p.218).

I have the following equation: $$\ddot{\delta} + 2H\dot{\delta} -\frac{3}{2} \Omega_m H^2 \delta = 0 $$

Then using the Fourier transform

$$\delta_{\vec{k}} = \frac{1}{V} \int \delta(\vec{r}) e^{i \vec{k} \cdot \vec{r}} d^3 r $$

Where $\delta(\vec{r})$ is the density fluctuation.

We get: $$\ddot{\delta_{\vec{k}}} + 2H\dot{\delta_{\vec{k}}} -\frac{3}{2} \Omega_m H^2 \delta_{\vec{k}} = 0 $$

The only function that does that is a gaussian function, I guess. I don't understand the process here.

Thank you