Wide ranges of engineering problems are significantly influenced by nonlinear equations. Therefore, variable approximate solution had been presented for solving these equations. In this paper, we apply the Laplace Homotopy Method to acquiring analytical solutions with variable coefficients. This technique does not require linearization or small perturbations. By using an initial value, the explicit solutions of the equation for different cases have been derived, which accelerate the rapid convergence of the series solution. Comparison with the exact solution shows that the LHM is a powerful method for the solution of linear and nonlinear differential equations. The Laplace Homotopy Method (LHM) is a promising method for handling functional equations. Some examples in different fields are given to illustrate that an appropriate choice of an initial solution can lead to the exact solution, these revealing the reliability and effectiveness of the method.

nonlinear problem, approximate solution, analytical solution, boundary values, fin.