Some results for a certain class of nonlinear binomial differential equation

In this paper, we study one class of nonlinear binomial differential equation

$$(f(z){)}^n+a(f^{(k)}(z){)}^n=p_1{e}^{{\lambda }_{1}z}+p_2{e}^{{\lambda }_{2}z},$$

where $n,k\in {\mathbb{N}}^{\mathbb{+}}$, $a$, $p_i$, and ${\lambda }_i$ are nonzero constants for $i=1,2$. Some results answer the previous conjecture given by Linkui Gao and Junyang Gao.