In many recent works, many authors have demonstrated the usefulness of fractional\r\ncalculus in the derivation of particular solutions of a significantly large number of\r\nlinear ordinary and partial differential equations of the second and higher orders. The\r\nmain objective of the present paper is to show how this simple fractional calculus\r\nmethod to the solutions of some families of fractional differential equations would\r\nlead naturally to several interesting consequences, which include (for example) a\r\ngeneralization of the classical Frobenius method. The methodology presented here is\r\nbased chiefly upon some general theorems on (explicit) particular solutions of some\r\nfamilies of fractional differential equations with the Laplace transform and the\r\nexpansion coefficients of binomial series.
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