In this article, we discuss the growth of solutions of the second-order nonhomogeneous linear differential equation
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$f^{{\prime \prime }}+A_{1}(z)e^{az}f^{{\prime }}+A_{0}(z)e^{bz}f=F$ ,
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where $a$ , $b$ are complex constants and $A_{j}(z)\not\equiv 0$ $(j=0,1)$ , and $F\not\equiv 0$ are entire functions such that $\max \{\rho (A_{j})\ (j=0,1),\rho (F)\}\char60 1$ . We also investigate the relationship between small functions and differential polynomials $g_{f}(z)=d_{2}f^{{\prime \prime }}+d_{1}f^{{\prime }}+d_{0}f$ , where $d_{0}(z),d_{1}(z),d_{2}(z)$ are entire functions that are not all equal to zero with $\rho (d_{j})\char60 1$ $(j=0,1,2)$ generated by solutions of the above equation.