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

We study the diferential equations w2+R(z)(w(k))2= Q(z), where R(z), Q(z) are nonzero rational functions. We prove (1) if the diferential equation w2+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) =C1/2cosα(z), where α(z) is a primitive of 1/R1/2(z) such that C1/2cosα(z) is a transcendental meromorphic function.(2) if the diferential equation w2+ 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) =C1/2cos(az + b), where a2k=1/A.