Cauchy Problem for Ordinary Differential Equations of Distributed Order

We consider the initial value problem for an ordinary differential equation with distributed order differentiation operator regarded as the integral of the Riemann–Liouville fractional derivative with respect to the order of differentiation over a Borel measure with compact support. We construct the fundamental solution to the equation in terms of the Wright function with distributed parameters and obtain the general representation for solutions to this equation. We prove the existence and uniqueness theorem for the Cauchy problem and express the solution in an explicit form.