Nonlinear Loewy factorizable algebraic ODEs and Hayman’s conjecture
- 44 Downloads
In this paper, we introduce certain n-th order nonlinear Loewy factorizable algebraic ordinary differential equations for the first time and study the growth of their meromorphic solutions in terms of the Nevanlinna characteristic function. It is shown that for generic cases all their meromorphic solutions are elliptic functions or their degenerations and hence their order of growth is at most two. Moreover, for the second order factorizable algebraic ODEs, all their meromorphic solutions (except for one case) are found explicitly. This allows us to show that a conjecture proposed by Hayman in 1996 holds for these second order ODEs.
Unable to display preview. Download preview PDF.
- A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Vestnik Moskovskogo Universiteta 1 (1937), 1–25.Google Scholar
- F. Schwarz, Loewy Decomposition of Linear Differential Equations, Texts and Monographs in Symbolic Computation, Springer, Vienna, 2012.Google Scholar