Meromorphic solution of a certain type of algebraic differential equation

Abstract

In the paper, we use the idea of normal family to find out the possible solution of the following special case of algebraic differential equation \[P_k\big(z,f,f^{(1)},\ldots, f^{(k)}\big)=f^{(1)}(f-\mathscr{L}_k(f))-\varphi (f-a)(f-b)=0,\] where $\mathscr{L}_k(f)=\sideset{}{_{i=0}^k}{\sum} a_i f^{(i)}$ and $\varphi$ is an entire function, $a_i\in\mathbb{C}\;(i=0,1,\ldots, k)$ such that $a_k=1$ and $a, b\in\mathbb{C}$ such that $a\neq b$. The obtained results improve and generalise the results of Li and Yang \cite{LY1} and Xu et al. \cite{XMD} in a large scale.