Semialgebraic integrability of algebraic differential equations

Abstract

Consider the algebraic differential equation P (z, w, w′ , w′′, . . . , w(n)) = 0, (1) where P is an irreducible polynomial in all variables. It was shown in [1] that if each particular solution of Eq. (1) satisfies some (in general, its own) algebraic differential equation of order less than n, then Eq. (1) is algebraically integrable, i.e., admits a first integral algebraic in all variables. Furthermore, algebraic differential equations can have first integrals algebraic in the arbitrary integration constant, the unknown function, and its derivatives and analytic in the independent variable. (In this case, one says that the general solution depends on the initial data of the Cauchy problem semitranscendentally; this term was used in [2, p. 427 of the Russian translation] for second-order equations without movable critical singular points.) For example, the equation w′′ = (2w + 1)w′ − w2 has the first integral w′ = w2 + Cez , where C is an arbitrary integration constant.