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In this paper, we introduce certain $n$-th order nonlinear Loewy factorizablealgebraic ordinary differential equations for the first time and study thegrowth of their meromorphic solutions in terms of the Nevanlinna characteristicfunction. It is shown that for generic cases all their meromorphic solutionsare elliptic functions or their degenerations and hence their order of growthare at most two. Moreover, for the second order factorizable algebraic ODEs,all the meromorphic solutions of them (except for one case) are foundexplicitly. This allows us to show that a conjecture proposed by Hayman in 1996holds for these second order ODEs.

Nonlinear Loewy Factorizable Algebraic ODEs and Hayman's Conjecture