J. Mech. Cont. & Math. Sci., Vol.-20, No.-2, February (2025) pp 115-127 Sabrina Sultana et al. 115 AN ADAPTED ANALYTICAL SOLUTION TO THE MULHOLLAND EQUATION: MODIFIED DIRECT ITERATION PROCEDURE Sabrina Sultana 1 , B M Ikramul Haque 2 , M. M. Ayub Hossain 3 1, 2 Department of Mathematics, Khulna University of Engineering & Technology, Khulna-9203, Bangladesh. 3 Department of Mathematics, Bangabandhu Sheikh Mujibur Rahman Science and Technology University, Pirojpur, Bangladesh. Email: 1 sabrinasultana1209@gmail.com, 2 ikramul@math.kuet.ac.bd 3 ayub.jpi@gmail.com Corresponding Author: M. M. Ayub Hossain https://doi.org/10.26782/jmcms.2025.02.00010 (Received: November 11, 2024; Revised: January 29, 2025; Accepted: February 11, 2025) Abstract The Mulholland equation is a third-order ordinary differential equation characterized by its nonlinearity. It particularly represents a central restoring force. Mulholland equations have many real applications in various fields, such as modern control theory, phase plane analysis, stability analysis, bifurcation analysis, etc. This paper employed the modified direct iteration method to solve the Mulholland equation, analytically incorporating all its terms (sometimes in approximated forms) during each iterative step. The analytical solutions are compared with existing results. The analytical solutions also showed remarkable precision when compared with numerical outcomes. It also becomes clear that compared to other approaches already in use, the modified direct iteration method is substantially easier to use, more accurate, efficient, and uncomplicated. Moreover, the fourth approximated frequency exhibited only a 0.022 percentage error. The suggested method can be widely applied to different engineering issues, while it is primarily demonstrated in nonlinear models with strong nonlinear factors. Keywords: Mulholland Equation, Direct iteration procedure, Analytical solution, Modified Direct iteration procedure, Mathematica. I. Introduction Nonlinear systems play a significant role in mathematics, medical science, life science, and other related fields. As a result, research on nonlinear systems has greatly benefited these areas. However, studying nonlinear systems is challenging and delicate JOURNAL OF MECHANICS OF CONTINUA AND MATHEMATICAL SCIENCES www.journalimcms.org ISSN (Online) : 2454 -7190 Vol.-20, No.-2, February (2025) pp 115 - 127 ISSN (Print) 0973-8975