Research Article An Effective Solution of the Cube-Root Truly Nonlinear Oscillator: Extended Iteration Procedure B. M. Ikramul Haque 1 and M. M. Ayub Hossain 1,2 1 Department of Mathematics, Khulna University of Engineering & Technology, Khulna 9203, Bangladesh 2 Department of Non-Technical, Shahid Abdur Rab Serniabat Textile Engineering College, Barishal, Bangladesh Correspondence should be addressed to B. M. Ikramul Haque; ikramul@math.kuet.ac.bd Received 23 June 2021; Revised 8 November 2021; Accepted 19 November 2021; Published 21 December 2021 Academic Editor: Sining Zheng Copyright © 2021 B. M. Ikramul Haque and M. M. Ayub Hossain. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e cube-root truly nonlinear oscillator and the inverse cube-root truly nonlinear oscillator are the most meaningful and classical nonlinear ordinary differential equations on behalf of its various applications in science and engineering. Especially, the oscillators are used widely in the study of elastic force, structural dynamics, and elliptic curve cryptography. In this paper, we have applied modified Mickens extended iteration method to solve the cube-root truly nonlinear oscillator, the inverse cube-root truly nonlinear oscillator, and the equation of pendulum. Comparison is made among iteration method, harmonic balance method, He’s amplitude-frequency formulation, He’s homotopy perturbation method, improved harmonic balance method, and homotopy perturbation method. After comparison, we analyze that modified Mickens extended iteration method is more accurate, effective, easy, and straightforward. Also, the comparison of the obtained analytical solutions with the numerical results represented an extraordinary accuracy. e percentage error for the fourth approximate frequency of cube-root truly nonlinear oscillator is 0.006 and the percentage error for the fourth approximate frequency of inverse cube-root truly nonlinear oscillator is 0.12. 1. Introduction Nonlinear systems are widespread around us. Nonlinear systems are widely involved in science, engineering, medical science, etc. So, research on the nonlinear systems has enriched science, engineering, medical science, etc. Research on nonlinear systems is complex and sensitive because most of the nonlinear systems suddenly change their character- istics due to some small changes of some valid parameters. e development of the theorems of dynamical systems has been derived by the modeling and formulating of nonlinear oscillators. us, nonlinear oscillators are one of the most important parts of nonlinear dynamical systems. Recently, many scientists have made significant improvement in finding a new mathematical tool which would be related to nonlinear dynamical systems whose understanding will rely not on analytic techniques but also on numerical and as- ymptotic methods. ey set up many fruitful and potential methods to operate the nonlinear systems. ere are many analytical methods to solve nonlinear oscillators such as perturbation [1–3]; standard and modified Poincar- e–Linstedt [4]; harmonic balance [5–7]; multiple scale [8]; homotopy perturbation [9–14]; modified He’s homotopy perturbation [15]; He’s frequency-amplitude formulation [16, 17]; cubication method [18, 19]; energy balance method [20]; He’s energy balance method [21]; Mickens iteration method [7, 22–25]; Hu’s iteration method [26]; Haque’s iteration method [27–29]; Haque’s extended iteration method [30–32]; variation iteration method [33]; homotopy analysis method [34]; finite element method and Akbari Ganji method [35]; and Taguchi method [36]. Besides, a few numbers of researchers have done work on the cube-root nonlinear oscillator using different methods such as the methods by Bel´ endez [9], Bel´ endez et al. [15], Ganji et al. [21], Mickens [25], Zhang [17], and Lim and Wu [6]. Among them, the proposed method is more easy, simple, and also Hindawi International Journal of Differential Equations Volume 2021, Article ID 7819209, 11 pages https://doi.org/10.1155/2021/7819209