Good day to you all,
This question is fairly open, so I hope that I'm not in breach of any of the Stack Exchange rules.
Clearly $f(t)=K_{\nu}(\beta t)$, where K is the modified Bessel function of the second kind, is a solution to Kelvin's equation. [Which is the modified Bessel equation with a small tweak]
I'm interested to know if anyone has come across a physically meaningful ODE (like Kelvin's equation \ Bessel equations etc) that has a solution of the form
\begin{equation} t^{\nu}K_{\nu}(\beta t) \end{equation}
Cheers, Allen.