Differential equation in spectral domain - meaning? Is there any occurrence of ODEs in spectral domain?
Let me mark the words: an ODE that applies to the Fourier transforms of functions describing a physical phenomena.
That is distinct from spectral theory that applies to ODEs in real domain, I believe. 
Edit : to be more specific, an equation of type 
$$ a\frac{d^2F(\omega)}{d\omega^2}+b\frac{dF(\omega)}{d\omega}+cF(\omega)=0 $$
$F(\omega)$ being the Fourier transform of $f(x)$.
 A: Here is a different answer in case you mean an ODE in the Fourier variables. If the ODE is to be, say, linear in the Fourier domain then its counterpart in the real domain must have coefficients that are polynomials in $x$ or $t$. I couldn't think of a situation where this occurs "naturally". Regardless, we can build our own situation. Consider waves propagating along an inhomogeneous beam. The hypothetical beam is made of a light material in the center and of increasingly heavy materials as you move away from the center say with a mass density that is quadratic in $x$. Then, the wave equation, in arbitrary units, can be $\partial_{x}^2 u = (1+x^2)\partial_{t}^2u$ for instance, or in Fourier domain: $-k^2u = -\omega^2u + \omega^2\partial_k^2 u$.
A: Here's one example. In the early universe, the density perturbations $\delta({{\bf x}, t}) = 1 - \rho({{\bf x}, t}) /\overline{\rho}(t)$ evolve according to the expression
$$
\frac{\partial^2\delta}{\partial t^2} + 2\frac{\dot{a}}{a}\frac{\partial \delta}{\partial t} = 4\pi G\overline{\rho}\delta + \frac{c_S^2}{a^2}\nabla^2\delta + \frac{2}{3}\frac{\overline{T}}{a^2}\nabla^2 S \tag{1}
$$
where $a$ is the scale factor, $\overline{T}$ the average temperature, $c_S$ the sound speed, and $S$ the entropy. It is convenient to write
$$
\delta({\bf x}, t)= \sum_{{\bf k}}\delta_{\bf k}(t)e^{i{\bf k}\cdot {\bf x}} \tag{2}
$$
which leads to
$$
\frac{{\rm d}^2\delta_{\bf k}}{{\rm d}t^2} + 2 \frac{\dot{a}}{a} \frac{{\rm d}\delta_{\bf k}}{{\rm d}t} = \left[4\pi G\overline{\rho} - \frac{k^2 c_S^2}{a^2}\right]\delta_{\bf k} - \frac{2}{3}\frac{\overline{T}}{a^2}k^2S_{\bf k} \tag{3}
$$
which has several advantages. One of them, is that in the linear regime, all modes are decoupled.

Application: Gravitational instability 

If the evolution of each mode is adiabatic, then $S_{\bf k} = 0$ and in a static universe Eq. (3) becomes
$$
\frac{{\rm d}^2\delta_{\bf k}}{{\rm d}t^2} = -\omega_{\bf k} \delta_{\bf k} ~~~\mbox{with}~~~ \omega_{\bf k} = \frac{k^2c_S^2}{a^2} - 4\pi G\overline{\rho} \tag{5}
$$
Define the characteristic length
$$
\lambda_J = \frac{2\pi a}{k_J} = c_S\left(\frac{\pi}{G\overline{\rho}}\right)^{1/2} \tag{6}
$$
which just represents the length traveled by a sound-wave in a gravitational free-fall time. For $\lambda < \lambda_J$ it is clear that $\omega^2_{\bf k} > 0$ and the solution oscillates, that is, for the amplitude of perturbation oscillate in time. However a different story happens for $\lambda > \lambda_J$ in this case this $\omega^2_{\bf k} < 0$ and the perturbation will grow with time.
And this is pretty much how galaxies form!
