Abstract— In this paper, differential calculus was used to obtain the ordinary differential equations (ODE) of the probability density function (PDF), Quantile function (QF), survival function (SF), inverse survival function (ISF), hazard function (HF) and reversed hazard function (RHF) of Kumaraswamy distribution. The parameters and support that define the distribution inevitably determine the nature, existence, uniqueness and solution of the ODEs. The method can be extended to other probability distributions, functions and can serve an alternative to estimation and approximation. Computer codes and programs can be developed and used for the implementation. Index Terms— Differentiation, quantile function, survival function, approximation, hazard function, Kumaraswamy. I. INTRODUCTION ALCULUS is a very key tool in the determination of mode of a given probability distribution and in estimation of parameters of probability distributions, amongst other uses. The method of maximum likelihood is an example. Differential equations often arise from the understanding and modeling of real life problems or some observed physical phenomena. Approximations of probability functions are one of the major areas of application of calculus and ordinary differential equations in mathematical statistics. The approximations are helpful in the recovery of the probability functions of complex distributions [1-10]. Apart from mode estimation, parameter estimation and approximation, probability density function (PDF) of probability distributions can be expressed as ODE whose solution is the PDF. Some of which are available. They include: beta distribution [11], Lomax distribution [12], beta prime distribution [13], Laplace distribution [14] and raised cosine distribution [15]. The aim of this research is to develop homogenous ordinary differential equations for the probability density function (PDF), Quantile function (QF), survival function (SF), inverse survival function (ISF), hazard function (HF) Manuscript received December 9, 2017; revised January 15, 2018. This work was sponsored by Covenant University, Ota, Nigeria. H. I. Okagbue, T. A. Anake and A. A. Opanuga are with the Department of Mathematics, Covenant University, Ota, Nigeria. hilary.okagbue@covenantuniversity.edu.ng abiodun.opanuga@covenantuniversity.edu.ng M. O. Adamu is with the Department of Mathematics, University of Lagos, Akoka, Lagos, Nigeria and reversed hazard function (RHF) of Kumaraswamy distribution. This will also help to provide the answers as to whether there are discrepancies between the support of the distribution and the necessary conditions for the existence of the ODEs. Similar results for other distributions have been proposed, see [16-28] for details. Kumaraswamy Distribution is one of the interval bounded support probability distributions and introduced by Kumaraswamy [29]. It is one of the most studied probability distribution as evidenced by the many research materials available. Some of the advantages of the distribution over the beta distribution were highlighted in [30] and [31]. A short note of the distribution was written by Nadarajah [32]. The boundary properties and inference of the distribution were discussed extensively in Okagbue [33] and Wang et al., [34] respectively. Some aspects of the distribution investigated by authors include: generalized order statistics [35], improved point estimation [36], Bayesian estimation of the parameters under censored samples [37-38], conditional estimation [39], analysis of the distribution based on record data [40] and statistical moments for the generalized distribution [41]. Its flexibility, ease of computation and tractability properties of the distribution are motivations for the numerous modifications and generalizations of the distribution. Some of which are listed as follows; exponentiated Kumarswamy distribution [42], Kumaraswamy Weibull distribution [43], bivariate Kumaraswamy distribution [44], Kumaraswamy generalized gamma distribution [45], Kumaraswamy Lindley distribution [46]. Also available are; Kumaraswamy- generalized Lomax distribution [47], Kumaraswamy Pareto distribution [48], Kumaraswamy-geometric distribution [49], Kumaraswamy Birnbaum–Saunders distribution [50], Kumaraswamy linear exponential distribution [51], Kumaraswamy-generalized exponentiated Pareto distribution [52], generalized Kumaraswamy exponential distribution [53], Kumaraswamy power series distribution [54] and [55]. Also available are; Kumaraswamy GP distribution [56], Exponentiated Kumaraswamy-Dagum distribution [57] and [58], Kumaraswamy Quasi Lindley distribution [59], transmuted Kumaraswamy distribution [60], exp-kumaraswamy distributions [61], the weighted kumaraswamy distribution [62], Kumaraswamy skew- normal distribution [63] Kumaraswamy modified inverse Classes of Ordinary Differential Equations Obtained for the Probability Functions of Kumaraswamy Distribution Hilary I. Okagbue, IAENG, Muminu O. Adamu, Timothy A. Anake and Abiodun A. Opanuga C Proceedings of the International MultiConference of Engineers and Computer Scientists 2018 Vol I IMECS 2018, March 14-16, 2018, Hong Kong ISBN: 978-988-14047-8-7 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) IMECS 2018