In this paper, we consider the problem of the exact and approximate solutions of certain differential equations with variational derivatives of the first and second orders. Some information about the variational derivatives and explicit formulas for the exact solutions of the simplest equations with the first variational derivatives are given. An interpolation method for solving ordinary differential equations with variational derivatives is demonstrated. The general scheme of an approximate solution of the Cauchy problem for nonlinear differential equations with variational derivatives of the first order, based on the use of the operator interpolation apparatus, is presented. The exact solution of the differential equation of the hyperbolic type with variational derivatives, similar to the classical Dalamber solution, is obtained. The Hermite interpolation problem with the conditions of coincidence in the nodes of the interpolated and interpolation functionals, as well as their variational derivatives of the first and second orders, is considered for functionals defined on the sets of differentiable functions. The found explicit representation of the solution of the given interpolation problem is based on an arbitrary Chebyshev system of functions. This solution is generalized for the case of interpolation of functionals on one out of two variables and applied to construct an approximate solution of the Cauchy problem for the differential equation of the hyperbolic type with variational derivatives. The description of the material is illustrated by numerous examples.
Some methods of solution to the Cauchy problem for a inhomogeneous equation of hyperbolic type with a Bessel operator
