Is there an encyclopedia of DE that accepts solutions?

You mean analytic solutions? There are only very few DE's with an analytic solution.

 

Here's a website that has general solutions to a number of common differential equations:

http://eqworld.ipmnet.ru/en/solutions/ode.htm

The page I linked to focuses on ordinary differential equations, but the menu on the left side of the page has links to systems of equations as well as PDEs.

 

eqworld is the best online source. I have the books of Kamke and Murphy, and although they contain a lot of de's with solutions and solution strategies, they are a bit dated. You should mainly use these as a database, not as study material. Much has happened since those books appeared: differential Galois theory, Lie's symmetry analysis, Darboux methods, to name a few. We know more about classification, relationships between DE's, equivalence transformations, etc.

 

I also have the book, but I am still in chapter 2. I think it is pretty much the state of affairs regarding Janet bases. I like the algorithmic approach, you can program the algorithm yourself which helps in understanding the method.
There are also people like van der Put and Singer who focus more on linear differential equations, their book is also on my To Do list. On my list is also the thesis of Aistleitner, which gives a nice overview and comparison of some other differential elimination methods
http://www.risc.jku.at/publications/download/risc_4229/master.pdf
I don't have a complete overview of the current state of affairs, this is all sunday-afternoon math for me, and I am happy to hear of other seminal 'must-read' works.

I don't know when mathematicians consider a mathematical approach to be complete. At least a lot of progress has been made the last 25 years and we can systematically find properties of large classes of differential equations (symmetries, invariant solutions, first integrals).

 

I think Lie groups are used in quantum mechanics, but I don't know a lot about it.