Теория дробного затухающего осцилляторного уравнения

General solution of the Cauchy problem for the class of fractional differential equations of the oscillatory type with attenuating part in the operator field of relations is found in the paper. For new generalized function of the Mittag-Leffler type with the help of which the general solution is represented a series of basic properties is being proved. Formal examples of the equation theory application in some generalized problems of theoretical mechanics such as motion of mathematical pendulum, motion of spherical pendulum, motion of heavy symmetric top with fixed low point and the Foucault pendulum theory are given.