Have you learned how to solve all ordinary differential equations?

Question

Mathematics

It is well known that not all partial differential equations can be solved (at least the same Schrödinger or Navier-Stokes equation). But the differential equations are "ordinary" - have they all been solved? Any kind? If so, where can I find how to classify the equation, and find a method for solving it. And if not, how can one "at a glance" determine whether the equation is solvable or not?

Answer 1

I don't know how everyone is, but I learned. book Matveev "Collection of tasks" - all basic types. at a glance - or from experience immediately the thought comes, or then by sorting through the methods of solution and types

Answer 2

Do you mean "solve analytically, find solution in the form of elementary functions"? Obviously, not all, because there are integrals that cannot be taken. For example, the equation y'(x)=sin(x)/x cannot be solved, we have to introduce a new function - the integral sine. As far as I remember, there is an algorithm for checking "whether a given integral is taken in a given set of functions" exists. For a general ODE 20 years ago, there was no such algorithm yet, whether it has appeared since then - I don’t know.

Answer 3

Not all are solved, of course (there are infinitely many of them theoretically). At a glance, it’s impossible to determine whether an equation can be solved analytically or not (you can only attribute it to a certain class, and only then can you see if they can solve such a class of equations or not).