Analytical solution for deflection of plate

I am a third yr undergraduate student of Aerospace Engineering department, IIT Kharagpur. I have a very simple question in strucutral mechanics. Is there any analytical solution available for deflection of plate under distributed load? The governing equation is well known and has been attached as an image file. Only classical plate theory can be considered. The boundary conditions can be taken as classical boundary conditions. So that can be any combination of a) hinged b) clamped and c) free boundary conditions.

If the solution is available, please suggest me the appropiate reference. If the solution is not available, please tell me whether any research is going on that direction.

I will welcome any appeal for clarification, required in this question.