All of the Pearson and Burr distributions are solutions to particular differential equations, of the pdf and CDF, respectively. Why was this a good strategy for generating distributions? Why do so many empirically useful distributions emerge from this approach? Aside from the "phase diagrams" that you sometimes see for selecting the right distribution based on the relative magnitude of the moments, is there anything about the particular differential equation associated with a given distribution that provides information about the use for which it is or can be good? The denominator of the RHS of the differential equation Pearson uses to define his system is a quadratic in x. I notice that when the first two coefficients of that quadratic are zero, the Normal distribution emerges. I assume this is not merely a happy accident.
Were Pearson and Burr simply seeking to create a range of continuous distributions that could match arbitrary combinations of moments, or is there something especially desirable about a distribution that matches a specified set or range of moments in this way?