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On the solvability and unsolvability of equations in explicit form. (English. Russian original) Zbl 1069.34133

Russ. Math. Surv. 59, No. 4, 661-736 (2004); translation from Usp. Mat. Nauk 59, No. 4, 69-146 (2004).
The survey is devoted to the solvability and unsolvability of equations in explicit form. The classical theory of Abel, Liouville, Galois, Picard, Vessiot, Kolchin and others is described and discussed. For example, the one-dimensional topological version of Galois theory and the solvability of linear differential equations by quadratures and the Picard-Vessiot theory is discussed in detail. The survey is an excellent summary of the actual existing methods.

MSC:

34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
34A05 Explicit solutions, first integrals of ordinary differential equations
12F10 Separable extensions, Galois theory
12H99 Differential and difference algebra
34A30 Linear ordinary differential equations and systems
34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
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