Journal of Mathematics Research; Vol. 13, No. 5; October 2021 ISSN 1916-9795 E-ISSN 1916-9809 Published by Canadian Center of Science and Education 5 Applying Differential Forms and the Generalized Sundman Transformations in Linearizing the Equation of Motion of a Free Particle in a Space of Constant Curvature Joel M. Orverem 1, 2 , Y. Haruna 2 , Bala M. Abdulhamid 2 and Magaji Y. Adamu 2 1 Department of Mathematical Sciences, Federal University Dutsin-Ma, Katsina State, Nigeria 2 Department of Mathematical Sciences, Abubakar Tafawa Balewa University Bauchi, Bauchi State, Nigeria Correspondence: Joel M. Orverem, Department of Mathematical Sciences, Federal University Dutsin-Ma, Katsina State, Nigeria. E-mail: orveremjoel@yahoo.com Received: June 23, 2021 Accepted: July 21, 2021 Online Published: August 25, 2021 doi:10.5539/jmr.v13n5p5 URL: https://doi.org/10.5539/jmr.v13n5p5 Abstract: Equation of motion of a free particle in a space of constant curvature applies to many fields, such as the fixed reduction of the second member of the Burgers classes, the study of fusion of pellets, equations of Yang-Baxter, the concept of univalent functions as well as spheres of gaseous stability to mention but a few. In this study, the authors want to examine the linearization of the said equation using both point and non-point transformation methods. As captured in the title, the methods under examination here are the differential forms (DF) and the generalized Sundman transformations (GST), which are point and non-point transformation methods respectively. The comparative analysis of the solutions obtained via the two linearizability methods is also taken into account. Keywords: differential equations, Differential Forms, Equation of Motion, free particle, generalized Sundman transformation, linearization, point and non-point transformation, second order, space of constant curvature 1. Introduction The equation of motion of a free particle in a space of constant curvature is a differential equation of second order. It finds applications in many areas as stated earlier. Such areas include spheres of gaseous stability, equations of Yang-Baxter, the study of fusion of pellets, the concept of univalent functions, and the static reduction of the second member of the Burgers classes, see (Nakpim & Meleshko, 2010) and (Karasu & Leach, 2009). The equation  ′′ + 3 ′ + 3 = 0, (1) was mentioned in (Mahomed, 2007) in line with the invertible symmetry group transformation, and the point transformation = 1  ,=+ 1  (2) was presented. The equation possesses the (3 ℝ) algebra of Lie point symmetries (Karasu & Leach, 2009). In other words, equation (1) admits the maximum eight-dimensional Lie algebra. The method of differential forms which was first investigated by (Harrison, 2002), was used earlier in (Orverem, Tyokyaa & Balami, 2017) and applied in (Orverem, Azuaba & Balami, 2017) to investigate the possibility of linearizing equation (1). The GST method was first established in (Duarte, Moreira & Santos, 1994) and later, in (Mustafa, Al-Dweik & Mara'beh, 2013) where only the Laguerre form was investigated. Equation (1) was also stated in (Nakpim & Meleshko, 2010) with a view of linearization through the generalized Sundman transformations (GST), but its solution was not obtained. This research is novel because the complete solution of equation (1) is obtained with the aid of the two methods under consideration DF and GST. Note that (Nakpim & Meleshko, 2010) presented the complete form of linearization through the GST. This method was also used by (Orverem, Haruna, Abdulhamid & Adamu, 2021) to linearized the essential Emden differential equation as the complete solution was obtained. In this present work, the authors apply the differential forms and the Sundman transformations methods in linearizing this