© 2023, IJSRMSS All Rights Reserved 9 International Journal of Scientific Research in Mathematical and Statistical Sciences Vol.10, Issue.2, pp.09-16, April 2023 E-ISSN: 2348-4519 Available online at: www.isroset.org Research Paper Methods for solving ordinary differential equations of second order with coefficients that is constant Shohal Hossain 1* , Samme Akter Mithy 2 1 Center for Multidisciplinary Research, Gono Bishwabidyalay, Dhaka-1344, Bangladesh 2 Center for Community Health and Research, Gonoshasthaya Samaj Vittik Medical College, Savar-1344, Dhaka, Bangladesh *Corresponding Author: sohel6944@gmail.com Received: 05/Mar/2023; Accepted: 04/Apr/2023; Published: 30/Apr/2023 Abstract— This article provides a detailed overview of the techniques and methods used to solve second-order ordinary differential equations having coefficients that remain constant. The article begins by introducing the general form of 2nd-order differential equations and explaining the concept of constant coefficients. Next, the article presents the characteristic of equation and its roots, which are used to determine the nature of the solutions. The article then goes on to discuss the three possible cases: distinct and real roots, complex conjugate roots, and repeated roots, and present the general solution for each case, including examples to illustrate the application of the method. The article concludes with a brief discussion of applications of second-order differential equations in engineering and physics. The aim of this article is to provide a clear and concise guide for students and researchers interested in this important topic. Keywords— Second-order differential equations, Constant coefficients, Homogeneous linear equations Real and distinct roots, Complex conjugate roots, Repeated roots 1. Introduction Ordinary differential equations of second order, whose coefficients remain constant throughout the equation are a class of equations that frequently arise in many branches of mathematics and physics, making them of fundamental importance in understanding and predicting physical phenomena. These equations have a wide range of applications in areas such as mechanics, electromagnetism, and quantum mechanics [1]. Therefore, finding the solutions to these equations is an essential task for many researchers and students in various fields of study. In this article, we aim to provide a comprehensive and detailed overview of the methods used to solve this class of equations. We begin by introducing the Standardized format of the second-order differential equation that has coefficients that remain constant throughout the equation, which is a homogeneous linear equation. We explain the meaning of constant coefficients and why they play an important role in solving these equations. We then present the characteristic equation and its roots, which provide information about the nature of the solutions. We discuss the three possible cases: real and distinct roots, complex conjugate roots, and repeated roots. For each case, we derive the general solution and provide examples to illustrate the application of the method [2]. One of the strengths of this article is that it covers not only the mathematical aspects of solving these equations but also their physical interpretations and applications. We discuss the connection between the solutions and physical systems and how the solutions can be used to analyse and predict the behaviour of these systems. We provide examples from physics and engineering to illustrate the applications of differential equation that has coefficients that remain constant throughout the equation [1] [2]. The techniques and methods presented in this article are fundamental to many fields of study, including mathematics, physics, and engineering. The article aims to provide a comprehensive guide for students and researchers who are interested in this topic, and it may be used as a reference for solving problems in related fields [1]. 2. Related Work One of the earliest works on this topic was by Leonhard Euler in the 18th century. Euler developed a general method for solving LDE (linear differential equations) of any order, including the 2 nd order equations coefficients with constant (Euler, 1748). This method involved finding the solutions or