The transcendental meromorphic solutions of a certain type of nonlinear differential equations

In this paper, we study the differential equations of the following form w2 + R(z)(w(k))2 = Q(z), where R(z), Q(z) are nonzero rational functions. We proved the following three conclusions: (1) If either P(z) or Q(z) is a nonconstant polynomial or k is an even integer, then the differential equation w2 +P(z)2(w(k))2 = Q(z) has no transcendental meromorphic solution; if P(z), Q(z) are constants and k is an odd integer, then the differential equation has only transcendental meromorphic solutions of the form f (z) = a cos(bz + c). (2) If either P(z) or Q(z) is a nonconstant polynomial or k >1, then the differential equation w2 +(z−z0)P (z)2(w(k))2 = Q(z) has no transcendental meromorphic solution, furthermore the differential equation w2 +A(z−z0)(w )2 = B, where A, B are nonzero constants, has only transcendental meromorphic solutions of the form f (z) = a cos b√z − z0, where a, b are constants such that Ab2 = 1, a2 = B. (3) If the differential equation w2 + 1 P(z)2 (w(k))2 = Q(z), where P is a nonconstant polynomial and Q is a nonzero rational function, has a transcendental meromorphic solution, then k is an odd integer and Q is a polynomial. Furthermore, if k = 1, then Q(z) ≡ C (constant) and the solution is of the form f (z) = B cos q(z), where B is a constant such that B2 = C and q (z)=±P(z). © 2007 Elsevier Inc. All rights reserved.