On a question of Gundersen-Yang concerning entire solutions of binomial differential equations

Abstract The explicit forms of entire solutions of the binomial differential equations a ( z ) f ′ f ″ − b ( z ) ( f ) 2 = c ( z ) e 2 d ( z ) , a ( z ) f f ′ − b ( z ) ( f ″ ) 2 = c ( z ) e 2 d ( z ) , $$a\left(z\right){f}^{\prime }{f}^{{\prime\prime}}-b\left(z\right){\left(f\right)}^{2}=c\left(z\right){e}^{2d\left(z\right)}, a\left(z\right)f{f}^{\prime }-b\left(z\right){\left({f}^{{\prime\prime}}\right)}^{2}=c\left(z\right){e}^{2d\left(z\right)},$$ are obtained by using the Nevanlinna theory, where a ( z ), b ( z ) and c ( z ) are polynomials such that a ( z ) b ( z ) c ( z ) ≢ 0, and d ( z ) is a non-constant polynomial. These results partially answer the question posed by Gundersen-Yang [Comput. Methods Funct. Theory. 21(2021), 605–617], and improving the previous results. In addition, some examples are given to illustrate these results.