On questions posed by Gundersen and Yang concerning a certain binomial differential equation

Suppose \(a, b, c\) are polynomials with \(ab \neq 0\) and \(c\) nonconstant. We determine all entire solutions to the equation \begin{equation*} f'f'' - a(z)f^2 = b(z)e^{2c(z)}. \end{equation*} This completely solves two open questions posed by Gundersen and Yang in [Comput. Methods Funct. Theory 21 (2021), 605-617].