On some overdetermined systems of three differential equations in second-order partial derivatives

This paper considers carefully a linear system of differential equations in partial derivatives from the deformable solid body mechanics and a class of systems of three equations including three second-order derivatives of one unknown function on the right-hand side in a non-linear manner. Using replacement of the first and second orders in the right parts with new unknown functions, transition to systems with a larger number of unknown functions has been performed and links with systems in total differentials have been established. For the studied overdetermined systems, the explicit compatibility conditions have been found and the variety of solutions containing not more than seven arbitrary constants have been determined. The obtained results, though mainly theoretical, can be used for further development of the theory of systems of differential equations with partial derivatives to solve specific problems of hydrodynamics (i.e., for construction of exact (“test”) solutions of the plane problems of hydrodynamics), gas dynamics, theory of elasticity, as well as a number of branches of mechanics, physics, and other basic sciences.