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

Abstract

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 polynomial or k is an even integer, then the differential equation w 2 + 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 ( b z + 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 z − z 0 , where a, b are constants such that A b 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 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 ) .