On the transcendental solutions of a certain type of nonlinear differential equations (Q1881576)

The authors study transcendental meromorphic solutions of differential equations of the form \[ f^{2}(z) + b(z)(L(f))^{2}=a(z), \] where \(L(f)\) denotes a linear differential polynomial in \(f\), \(a(z)\) and \(b(z)\) are nonzero small functions of \(f(z)\). It turns out that if all coefficients of the above equation are in \(C[z]\), then such solution has the very special form \[ f(z)=P(z)\exp{R(z)} + Q(z)\exp\{-R(z)\}. \] This result is obtained by using Nevanlinna's value distribution theory. Reviewer's remark: Lemma 3 is not formulated exactly. Its statement is true in one direction only.