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...$