The transcendental meromorphic solutions of a certain type of nonlinear differential equations (Q2371866)

The paper concerns differential equations of the form \[ w^2+R(z)(w^{(k)})^2 = Q(z), \] where \(R, Q\) are rational functions. The authors give a detailed discussion and study of the meromorphic solutions of the equation. For example, they show that if either \(R(z)\) or \(Q(z)\) is a nonconstant polynomial or \(k\) is an even integer, then the equation has no meromorphic solution; if \(R(z)\) and \(Q(z)\) are constants and \(k\) is an odd integer, then the equation has only transcendental meromorphic solutions of the form \(f(z)= a \cos (bz+c)\).