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In this paper, we investigate the relationship between small functions and differential polynomials $g_f\left(z\right)=d_2{f}^{\text{'}\text{'}}+d_1{f}^{\text{'}}+d_{0}f$, where $d_0\left(z\right)$, $d_1\left(z\right)$, $d_2\left(z\right)$ are entire functions that are not all equal to zero with $\rho \left(d_j\right)<1$ $(j=0,1,2)$ generated by solutions of the differential equation $f^{\text{'}\text{'}}+A_1\left(z\right)e^{az}{f}^{\text{'}}+A_0\left(z\right)e^{bz}f=F$, where $a,b$ are complex numbers that satisfy $ab(a-b)\ne 0$ and $A_j\left(z\right)\neg \equiv 0$ ($j=0,1$), $F\left(z\right)\neg \equiv 0$ are entire functions such that $max\left\{\rho \left(A_j\right),j=0,1,\rho \left(F\right)\right\}<1.$

In this paper we discuss the growth of solutions of the higher order nonhomogeneous linear differential equation $$\begin{array}{cc}& f^{\left(k\right)}+A_{k-1}{f}^{(k-1)}+\cdots +A_2{f}^{\text{'}\text{'}}+(D_1\left(z\right)+A_1\left(z\right){\mathrm{e}}^{az})f^{\text{'}}\hfill \\ & +(D_0\left(z\right)+A_0\left(z\right){\mathrm{e}}^{bz})f=F(k\ge 2),\hfill \end{array}$$ where $a$, $b$ are complex constants that satisfy $ab(a-b)\ne 0$ and $A_j\left(z\right)$ $(j=0,1,\cdots ,k-1)$, $D_j\left(z\right)$ $(j=0,1)$, $F\left(z\right)$ are entire functions with $max\{\rho \left(A_j\right)(j=0,1,\cdots ,k-1),\rho \left(D_j\right)$ $(j=0,1)\}<1$. We also investigate the relationship between small functions and the solutions of the above equation.