We propose a general method to obtain the representation of solutions for linear fractional order differential equations based on the theory of (a, k)-regularized families of operators. We illustrate the method in case of the fractional order differential equation Dα t u (t) + µDα t u(t) = Au(t) + t−α Γ(1 − α) (u (0) + µu(0)) + f(t), t > 0, 0 < α ≤ 1, where A is an unbounded closed operator defined on a Banach space X and f is a X-valued function.