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

In this paper, we study the differential equations of the following form w(2) + 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 polynornial or k is an even integer, then the differential equation w(2) + P(z)(2)(w((k)))(2) = Q(z) has no transcendental rnerornorphic 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 w(2) + (z - z(0)) P(z)(2)(w((k)))(2) = Q(z) has no transcendental meromorphic, solution, furthermore the differential equation w(2) + A(z - z(0))(w ')(2) = B, where A, B are nonzero constants, has only transcendental meromorphic solutions of the form f (z) a cos b root z - z(0), where a, b are constants such that Ab(2) = 1, a(2) = B. (3) If the differential equation w(2) + 1/P(z)(2) (w((k)))(2)= Q(z), where P is a nonconstant polynomial T, (,,-)-2 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 B-2 = C and q '(z) = +/- P(z). (c) 2007 Elsevier Inc. All rights reserved.