This paper is devoted to studying the growth and the oscillation ofsolutions of the second order non-homogeneous linear differential equation\begin{equation*}f^{\prime \prime }+A_{1}\left( z\right) e^{P\left( z\right) }f^{\prime}+A_{0}\left( z\right) e^{Q\left( z\right) }f=F,\end{equation*}%where $P\left( z\right)$, $Q\left( z\right) $ are nonconstantpolynomials such that $\deg P=\deg Q=n$ and $A_{j}\left( z\right) $\linebreak $\left( \not\equiv 0\right) $ $(j=0,1),$ $F\left( z\right) $ areentire functions with $\max \{\rho \left( A_{j}\right) :j=0,1\}<n$. We alsoinvestigate the relationship between small functions and differentialpolynomials $g_{f}\left( z\right) \linebreak =d_{2}f^{\prime \prime }+d_{1}f^{\prime}+d_{0}f$, where $d_{0}\left( z\right) ,d_{1}\left( z\right) ,d_{2}\left(z\right) $ are entire functions such that at least one of $d_{0},d_{1},d_{2}$%\ does not vanish identically with $\rho \left( d_{j}\right) <n$% $\left( j=0,1,2\right) $ generated by solutions of the aboveequation.