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Can anyone point me to some references (preferably online) that would address the following kinds of questions:
What classes of differential equations are proven to be analytically solvable?
What classes of differential equations are proven to be analytically unsolvable?
For which classes of differential equations are there numerical algorithms for which the error term can be bounded by a definite function of the "epsilons" involved in the algorithm and the precision with which calculations are made?
Background: I'm writing a blog article on the rate equation, which includes a "fireside chat" with the reader about differential equations in general; what are their prospects for being solved analytically, and what are their prospects for being reasonably approximated by numerical methods. If we just take a general reaction network, write the rate equation for it, and apply the Euler method to it, then how can we be sure that, with a given machine precision, we know what delta-T will bring the error within a given tolerance.
I saw a lot of talk on the web about unsolvable differential equations, but I didn't find any clear explanations about which ones where proven to be unsolvable, and how they were so proven. On web page stated that all ordinary differential equations were solvable, whereas even some linear partial differential equations are unsolvable. Really? I was skeptical of this statement, also because it framed ordinary differential equations in the special form:
f'(t) = g(t,f(t))
But what about ordinary differential equations that don't express the derivative as a function of all the other terms?
Is there something like the differential Galois theory that is used to prove which integrals are "impossible," which applies to differential equations?
In going through this writing exercise, I learned that these are the basic questions that I need to learn some more about myself.
If anyone can offer clarifications here, that would also be very helpful.
Thanks very much.