Entire Solutions of Certain Type Binomial Differential Equations

Inspired by the questions Gundersen and Yang proposed, we investigate the exact forms of the entire solutions of the following two types of binomial differential equations a(z)ff ''+b(z)(f ')2=c(z)e2q(z);a(z)f ' f ''+b(z)f2=c(z)e2q(z),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} a(z)ff''+b(z)(f')<^>2=c(z)e<^>{2q(z)}; \\ a(z)f'f''+b(z)f<^>2=c(z)e<^>{2q(z)}, \end{aligned}$$\end{document}where a, b, c are polynomials with no common zeros satisfying abc not equivalent to 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$abc\not \equiv 0$$\end{document}, and q is a non-constant polynomial.