About one Briot–Bouqet equation

Abstract

This article is devoted to the problem of studying meromorphic solutions of algebraic differential equations, which is traditional for the theory of differential equations. At the present, the case of the linear equations is quite well explored. Speaking of the nonlinear equations, there are relatively few results related to more or less common equation classes. There is one class of equations, where a number of results have been obtained. They are called Briot–Bouquet equations. These are the equations of the form 𝑃(𝑦, 𝑦(𝑛)) = 0, where 𝑃 is a complex polynomial, 𝑛 ∈ N. The research of the meromorphic solutions of this type of equations was started by Ch. Briot, J. C. Bouqet and Ch. Hermit, who described all possible solutions of the equations of the form 𝑃(𝑦, 𝑦′) = 0 by showing that they are all included in class 𝑊, which consists of rational
functions, rational functions of some exponential function and elliptic functions. After that E.
Picard’s work was published where he proved that all solutions of the equations of the form 𝑃(𝑦, 𝑦′′) = 0 are also included in 𝑊.
Later, the hypothesis arose that in any 𝑃(𝑦, 𝑦(𝑛)) = 0 equation (with some limitations to the 𝑃) all its meromorphic solutions are included in 𝑊. E. Hille, R. Kaufman, S. Bank, A. Eremenko, L. Liao, T. Ng, A. Yanchenko and other mathematicians have been working on its proof. Nowadays the validity of the hypothesis has been established in many cases, but there are a number of cases left, where it is neither proved nor disproved.
There is one of these cases described in this work. Exactly, equations 𝑦(𝑛) = 𝑦𝑚, where 𝑛,𝑚 ∈ N, 𝑚 ⩾ 2. A necessary and sufficient condition for the existence of nonzero meromorphic solutions of these equations and these solutions themselves are found.