Solution of inhomogeneous systems for differential equations in private derivatives of the third order

Abstract

The possibilities of constructing inhomogeneous system solutions for partial differential equations of the third order have been studied. The construction of general and particular solutions corresponding to homogeneous system comprehensively investigated by using Frobenius-Latysheva method. Type of solutions near the special curves are established. The number of linearly independent partial solutions is determined. A theorem on the representation a general solution of inhomogeneous system is proved, and the application of uncertain coefficients method for such systems is revealed. On a concrete example, it is shown that the particular solutions of the inhomogeneous system constructed in this way are solutions of one inhomogeneous third - order equation obtained by adding the two equations of the considered example. One of particular solutions corresponding to homogeneous system relates to degenerate generalized hypergeometric series of Clausen type with two variables. Properties of generalized hypergeometric series are still poorly understood.