Abstract:
The construction of the structure theory of second order
scalar differential equations in the plane is considered.
Because of this we introduce the two main classes of
partial differential equations which are the subject of this
dissertation. The classes are the determined first order
systems of partial differential equations for two unknown
functions of two variables and second order scalar partial
differential equations in the plane. These two classes of
equation have similar geometric structure, and this is
precisely the reason that we can develop the theory for
these systems in parallel. Therefore, we investigate a
formulation of the structure theory of second order scalar
differential equations in the plane in terms of contact
distribution. In this research we consider the work of
vessiot that explain the geometric characterization of
these equations which summarized by Stomark and
Duistermaat, to discuss a new formulation and extend the
structure theory given by this authers, and then we
construct a framing on the equation manifold of first
4order systems that are invariant under general contact transformation as an application of the theory .
