Classes of Ordinary Differential Equations Obtained for the Probability Functions of Cauchy, Standard Cauchy and Log-Cauchy Distributions

In this paper, the differential calculus was used to obtain some classes of ordinary differential equations (ODE) for the probability density function, quantile function, survival function and hazard function of Cauchy, standard Cauchy and log-Cauchy distributions. The distributions are related by logarithmic transformation. The ODEs for the inverse survival function and reversed hazard functions of the distributions were not considered because of their nature and complexity. The stated necessary conditions required for the existence of the ODEs are consistent with the various parameters that defined the distributions. Solutions of these ODEs by using numerous available methods are a new ways of understanding the nature of the probability functions that characterize the distribution.