Initial-value problem for a second-order ordinary differential equation with distributed-order differentiation operator

We study a linear ordinary differential equation of the second order with operator of continuously distributed differentiation with variable coeficients. Such equations belong to the class of continuous differential equations. In this paper, the continuously distributed differentiation operator is defined as an integral with summable kernel from the Riemann-Liouville fractional di erentiation operator in the order of differentiation. A special case of the operator of continuously distributed differentiation is the operator of discretely distributed differentiation. For the equation under consideration, a fundamental solution is constructed in the form of a Neumann series, reducing the differential problem to the Volterra integral equation of the second kind, which is solved by the method of sequential Picard approximation. The qualitative and structural properties of the fundamental solution are proved with the help of which the solution to the Cauchy problem is found in terms of the fundamental solution using the Lagrange formula.