On entire solutions of a class of second-order algebraic differential equations (Q2049314)

Let \(\mathbb{C}(z)\) be the field of rational functions over \(\mathbb{C}\), and let \(\mathbb{C}[\omega_0,\ldots,\omega_n]\) denote the set of polynomials of variables \(\omega_0,\ldots,\omega_n\) over the field of complex numbers. Furthermore, let \(b_0,b_1,b_2\in\mathbb{C}[z]\) with \(b_2\not\equiv 0\) and let \(A\in\mathbb{C}[z,\omega_0,\omega_1,\omega_2]\) such that \(A[z,0,\omega_1,-(b_1/b_2)\omega_1]\not\in\mathbb{C}(z)\). In the paper under review it is shown that if \(y=f(z)\) is a finite-order entire solution of \[ b_2 y'' + b_1 y' + b_0y + y A(z,y,y',y'') =0, \] then there exist \(A_1,A_2\in\mathbb{C}[z]\) and \(B_1,B_2\in\mathbb{C}(z)\) such that \[ f(z)=B_1 e^{A_1} + B_2 e^{A_2}. \]