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.