Classes of Ordinary Differential Equations Obtained for the Probability Functions of Exponential and Truncated Exponential 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, inverse survival function and reversed hazard function of the exponential and truncated exponential distributions. However, the ODE of the hazard function of the exponential distribution cannot be obtained because of the constant failure nature of the distribution. The ODE of the truncated exponential distribution was obtained. 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 new ways of understanding the nature of the probability functions that characterize the distribution.