Fourier transform of a 6th order ODE

Question

How can I take the Fourier transform of a $6^\text{th}$ order ODE, which includes terms such as $\tfrac{1}{r}D^6 U(r)$ or $\tfrac{1}{r}D^3 U(r)$? Here $r$ is the independent variable and $U(r)$ is to be obtained.

Answer 1

In view of possible singularity of $U(r)$ at $r=0$ the Fourier transform of the ODE under consideration is not applicable in this case. It is possible that this transform does not exist. See Wiki for info.