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 \(a^{2k} = \tfrac{1} {A}\).