This research focuses on solving the nonlinear second-order jet engine vibration equation utilizing a hybrid analytical technique named the least square homotopy perturbation method (LSHPM). The numerical and graphical comparison of the solutions obtained using the homotopy perturbation method (HPM), LSHPM, and the MATLAB built-in solver bvp5c is presented across four distinct cases. Additionally, a comparative analysis between the solutions derived from LSHPM and those reported in previous literature is also presented. The tabular and graphical representation of the solutions, along with the numerical validation through residual error analysis, are given. Furthermore, the convergence analysis of the LSHPM for its stability and solution reliability is provided. The graphical and numerical representation of the residual error analysis reveals that LSHPM exhibits superiority over HPM in terms of rapid convergence and accuracy. The strong agreement between the results obtained from HPM and bvp5c with those of LSHPM demonstrates that LSHPM offers a more efficient, reliable, and fast convergent solution of the initial and boundary value problem.