The aim of this paper is to consider the growth of solutions of certain second-order linear differential equation. The coefficients of the equation are polynomials in the complex exponential function, while the coefficients of the polynomials are transcendental integral functions. The value distribution theory is mainly used to show that the hyper-order of every nontrivial solution of the equation equals one when the coefficients of the equation satisfy certain conditions. The proof can be divided into two steps: firstly, it is shown that the growth order of every nontrivial solution of the considered equation equals infinity by contradiction and the properties of transcendental meromorphic functions; secondly, it is shown that the hyper-order of every nontrivial solution of the equation equals one by contradiction and the Wiman-Valiron theory. The obtained results generalize some previous results.