Is there an encyclopedia of DE that accepts solutions?

[jonjacson]

Main Question or Discussion Point

DE= Differential equations.

There is an Encyclopedia of Integer Sequences for example, but I am not able to find the equivalent for DE.

I would like to find a list with all the differential equations that have been solved up to date.

A website or any other source would be interesting. If the source contains the methods to get their solutions would be perfect.

Thanks for your time.

 

[Mark44]

DE= Differential equations.

There is an Encyclopedia of Integer Sequences for example, but I am not able to find the equivalent for DE.

I would like to find a list with all the differential equations that have been solved up to date.

I doubt very much that such a list exists.

A website or any other source would be interesting. If the source contains the methods to get their solutions would be perfect.

Thanks for your time.

 

[jonjacson]

I doubt very much that such a list exists.

Maybe if we talk about a specific type of differential equations?

 

[micromass]

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

 

[jonjacson]

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

Yes, I mean that. Sorry for not expressing it properly maybe.

 

[TeethWhitener]

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.

 

[jonjacson]

That is what I was looking for. I will purchase those books.

Hopefully they will show the methods employed to find the solutions, I mean methods like the Euler method etc.

 

[bigfooted]

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.

 

[jonjacson]

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 have the book "ALgorithmic Lie Theory" that talks about another approach to transfering the Galois method to differential equations that was started by Picard and Vessiot. Do you know what is the "state of affairs" on this topic?

Apparently there are several approaches to apply Galois theory to differntial equations, I don't how if that has been completed succesfully and completely.

 

[bigfooted]

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).

 

[jonjacson]

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).

Very interesting.

Do you know if these methods are useful with quantum mechanics equations (Schrodinger)?

 

[bigfooted]

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