On the hyper-order of transcendental meromorphic solutions of certain higher order linear differential equations

In this paper, we investigate the growth of meromorphic solutions of the linear differential equation \[f^{(k)}+h_{k-1}(z)e^{P_{k-1}(z)}f^{(k-1)}+\ldots +h_{0}(z)e^{P_{0}(z)}f=0,\] where \(k\geq 2\) is an integer, \(P_{j}(z)\) (\(j=0,1,\ldots ,k-1\)) are nonconstant polynomials and \(h_{j}(z)\) are meromorphic functions. Under some conditions, we determine the hyper-order of these solutions. We also consider nonhomogeneous linear differential equations