Entire solutions of a class of binomial differential equations (Q6592725)

This paper addresses questions posed by \textit{G. G. Gundersen} and \textit{C.-C. Yang} [Comput. Methods Funct. Theory 21, No. 4, 605--617 (2021; Zbl 1502.34100)] regarding entire solutions of certain second-order binomial differential equations. Specifically, it examines the equation \( ff' - a(z)(f'')^2 = b(z)e^{2c(z)} \), where \( a(z) \), \( b(z) \), and \( c(z) \) are polynomials. The authors provide explicit forms of entire solutions under the condition that all zeros of \( f' \) are simple. The main results classify solutions according to the nature of the polynomials \( a(z) \), \( b(z) \), and \( c(z) \), providing conditions for when solutions can be expressed in terms of exponential functions multiplied by polynomials. Several examples are presented to illustrate the results.