  In this paper, for a transcendental meromorphic function $f$ and $a\in
\mathbb{C}$, we have exhaustively studied the nature and form of solutions of a
new type of non-linear differential equation of the following form which has
never been investigated earlier: \beas f^n+af^{n-2}f'+ P_d(z,f) =
\sum_{i=1}^{k}p_i(z)e^{\alpha_i(z)},\eeas where $P_d(z,f)$ is differential
polynomial of $f$, $p_i$'s and $\alpha_{i}$'s are non-vanishing rational
functions and non-constant polynomials respectively. When $a=0$, we have
pointed out a major lacuna in a recent result of Xue [Math. Slovaca,
70(1)(2020), 87-94] and rectifying the result, presented the corrected form of
the same at a large extent. The case $a\neq 0$ has also been manipulated to
determine the form of the solutions. We also illustrate a handful number of
examples for showing the accuracy of our results.
