In this paper, we investigate the order of growth of solutions of the higher order non-homogeneous linear differential equation \[ f^{(k)}+\sum_{j=0}^{k-1}h_je^{P_j(z)}f^{(j)}=F, \] where $P_j(z)(j=0,1,\cdots,k-1)$ are polynomials with deg $P_j=n_j\geq1$ and $h_j(z)(j=0,1,\cdots,k-1)$ not all vanishing identically, $F$ are meromorphic functions of finite order having only finitely many poles. Under some conditions, we prove that every meromorphic solution $f\not\equiv0$ of the above equation is of infinite order. We give also some estimates of their hyper-order, exponent of convergence of the zeros and the hyper-exponent of convergence of zeros. Furthermore, we give an estimation for the exponent of convergence of fixed points of solutions and their 1st, 2nd derivatives.