In the theory of complex analytic functions, the normality criterion is an important part of the families of meromorphic functions. In $1907$, Paul Montel's first introduced the notion of normal families. In the sense of Montel's, let $\Omega$ be a domain in $\mathbb{C}$ and let $\digamma$ be a family of holomorphic functions. The family $\digamma$ is said to be normal in $\Omega$ if every sequence $\{\xi_{n}\}\subseteq \digamma$ contains either a subsequence which converges to a limit function $\xi(\not\equiv \infty)$ uniformly on each compact subset of $\Omega$, or a subsequence which converge uniformly to $\infty$ on each compact subset.

In this paper, we study the uniqueness of a transcendental meromorphic function contributing to a meromorphic function in sync with its first derivative and a linear differential polynomial of first order with two constant coefficients using the theory of normal families. Our result generalizes and supplements some previous results given by Jank-Mues-Volkmann \cite{JMV1986}, Chang-Fang \cite{CF2002}, Chang \cite{C2003}, Lahiri-Ghosh \cite{LG2009}, L$\ddot{u}$-Yi \cite{LY2010} and L$\ddot{u}$-Xu \cite{LX2012}. We also provide examples to demonstrate the correctness of our results.