On a certain type of nonlinear differential equations admitting transcendental meromorphic solutions

Abstract

We study the differential equations w ²+R(z)(w ^(k))² = Q(z), where R(z),Q(z) are nonzero rational functions. We prove

• (1) if the differential equation w ²+R(z)(w′)² = Q(z), where R(z), Q(z) are nonzero rational functions, admits a transcendental meromorphic solution f, then Q ≡ C (constant), the multiplicities of the zeros of R(z) are no greater than 2 and f(z) = √C cos α(z), where α(z) is a primitive of $\tfrac{1} {{\sqrt {R(z)} }}$ such that √C cos α(z) is a transcendental meromorphic function.

• (2) if the differential equation w ² + R(z)(w ^(k))² = Q(z), where k ⩾ 2 is an integer and R,Q are nonzero rational functions, admits a transcendental meromorphic solution f, then k is an odd integer, Q ≡ C (constant), R(z) ≡ A (constant) and f(z) = √C cos (az + b), where $a^{2k} = \tfrac{1} {A}$.