Just a shot in the dark here, but what about this:
Thanks for the suggestion...but then again: Integrating w.r.t z is also a problem I wouldn't know how to solve..
The actual thing I'm trying to show here, is that not all solutions to this equation are bounded..., the question asks to solve the eq. explicitly
But I've no clue how. However, we may be able to use that for all t..., then try to show that all solution except the origin are unbounded.
(edit: i meant to show that not all solutions to this problem are bounded)
Yeah, I was thinking of a substitution somewhat like that, but I ran into difficulties. That one is better, I think. Query: does your solution satisfy the original DE?
I'm not so sure that this generates only bounded solutions. You don't know that the exponent there is purely imaginary, do you?