In this paper, the differential calculus (product
rule) was used to obtain some classes of ordinary differential
equations (ODE) for the probability density function, quantile
function, survival function, inverse survival function, hazard
function and reversed hazard function of the half-Cauchy and
power Cauchy distributions. 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 distributions. The method can be
extended to other probability distributions and can serve an
alternative to approximation especially the cases of the
quantile and inverse survival function
