Nonhomogeneous linear differential equations can often be solved by Fourier transforming the differential equation into an algebraic equation which is easier to solve. However, this only gives one particular solution, while the general solution of a linear differential equation forms an affine space of the schematic form (particular solution) + (all solutions to homogeneous part). The usual story I've heard to explain this discrepancy is that typically only one particular solution is Lebesgue-integrable and therefore has as well-defined Fourier transform, and only this solution can be found by the Fourier transform method. (Fourier transforming a homogeneous differential equation typically just yields the trivial zero solution, or possibly a set of plane waves with discrete wave vectors.)
Is it true that linear differential equations generically only have one integrable solution? That's not at all obvious to me. If so, is there an easy way to see why? If not, which particular solution does the Fourier transform method find?