2 00 4 Qualitative investigation of the solutions to differential equations ( Application of the skew - symmetric differential forms )

The presented method of investigating the solutions to differential equations is not new. Such an approach was developed by Cartan [1] in his analysis of the integrability of differential equations. Here this approach is outlined to demonstrate the role of skew-symmetric differential forms. The role of skew-symmetric differential forms in a qualitative investigation of the solutions to differential equations is conditioned by the fact that the mathematical apparatus of these forms enables one to determine the conditions of consistency for various elements of differential equations or for the system of differential equations. This enables one, for example, to define the consistency of the partial derivatives in the partial differential equations, the consistency of the differential equations in the system of differential equations, the conju-gacy of the function derivatives and of the initial data derivatives in ordinary differential equations and so on. The functional properties of the solutions to differential equations are just depend on whether or not the conjugacy conditions are satisfied. The basic idea of the qualitative investigation of the solutions to differential equations can be clarified by the example of the first-order partial differential equation. (1) be the partial differential equation of the first order. Let us consider the functional relation du = θ (2) where θ = p i dx i (the summation over repeated indices is implied). Here θ = p i dx i is the differential form of the first degree. The specific feature of functional relation (2) is that in the general case this relation turns out to be nonidentical. The left-hand side of this relation involves a differential, and the right-hand side includes the differential form θ = p i dx i. For this relation to be identical, the differential form θ = p i dx i must be a differential as well (like the left-hand side of relation (2)), that is, it has to be a closed exterior differential form. To do this it requires the commutator K ij = ∂p j /∂x i − ∂p i /∂x j of the differential form θ has to vanish.