List of ODE's that can be solved by Fourier transform

Question

I am teaching introductory level Fourier analysis and I want to give my students some basic and some not so basic examples of how to solve ordinary differential equations with the method of Fourier transform. Looking at various sources there seems to be basically just one instance $u''-u=f$ where the method is applicable (the other examples seem to be derived from it. Since there also other methods to deal with these examples and the students may be unconvinced what they actually gain by using this method, I wonder where could one find a list of ODE's (maybe with non-constant coefficients but not PDE's) which are solvable (at least in principle) precisely by the method of Fourier transform (not Laplace transform, Fourier series etc., I know these are related, yet my question is a particular one). Preferably the list should contain cases when distribution theory is either needed or is avoided.