On the growth of real solutions of second-order algebraic differential equations

Abstract

is considered. F* is supposed to be a polynomial with respect to J, J’, y”, whose coefficients are polynomials in x or continuous functions on [0, co) having some restrictions. Borel’s conjecture on the growth of solutions y(x) defined on [O, co) of algebraic equations F(x, JJ, y’,..., I,(~‘) = 0 was that 4’ = O(exp, Ax”) (exp, a = exp(exp,-, a}, exp, a = exp a, for n > 2), with some constants A and m. In [ 1 ], Borel’s conjecture was proved under some conditions imposed on (*). The conjecture was proved under some restrictions for a real solution J!(X) of (*) on a ray [x0, co) for which In J(X) is a convex function of In x. In [3], Bank proved Borel’s conjecture for increasing solutions J$X) which satisfy the additional condition J”/X”J -+ co, where a is a certain constant dependent on (*). In [ 21, Borel’s conjecture for a real solution J’(X), J(X) > a > 0 was proved under some restrictions imposed on Eq. (*). Such problems were investigated in [4-61. There are examples showing that Borel’s conjecture is in general wrong for IZ > 2. In particular Vijayraghavan [ 71 showed that there exist second-order algebraic differential equations whose positive solutions on [0, co) have no majorants expressed in terms of coefficients. In view of this result it seems reasonable to put some restrictions on the equation F*(x, J’, y’, 4”‘) = 0 and on the type of solutions in order to obtain the majorant expressed in terms of the coefficients of the equation.