Investigating the correspondence between systems of partial differential equations and their
analytic solutions using a formal approach this monograph presents algorithms to determine the
set of analytic solutions of such a system and conversely to find differential equations whose
set of solutions coincides with a given parametrized set of analytic functions. After giving a
detailed introduction to Janet bases and Thomas decomposition the problem of finding an
implicit description of certain sets of analytic functions in terms of differential equations
is addressed. Effective methods of varying generality are developed to solve the differential
elimination problems that arise in this context. In particular it is demonstrated how the
symbolic solution of partial differential equations profits from the study of the
implicitization problem. For instance certain families of exact solutions of the Navier-Stokes
equations can be computed.
