  We introduce methods for deriving analytic solutions from
differential-algebraic systems of equations (DAEs), as well as methods for
deriving governing equations for analytic characterization which is currently
limited to very small systems as it is carried out by hand. Analytic solutions
to the system and analytic characterization through governing equations provide
insights into the behaviors of DAEs as well as the parametric regions of
operation for each potential behavior. For each system (DAEs), and choice of
dependent variable, there is a corresponding governing equation which is
univariate ODE or PDE that is typically higher order than the constitutive
equations of the system. We first introduce a direct formulation for
representing systems of linear DAEs. Unlike state space formulations, our
formulation follows very directly from the system of constitutive equations
without the need for introducing state variables or singular matrices. Using
this formulation for the system of constitutive equations (DAEs), we develop
methods for deriving analytic expressions for the full solution (complementary
and particular) for all dependent variables of systems that consist of constant
coefficient ordinary-DAEs and special cases of partial-DAEs. We also develop
methods for deriving the governing equation for a chosen dependent variable for
the constant coefficient ordinary-DAEs and partial-DAEs as well as special
cases of variable coefficient DAEs. The methods can be automated with symbolic
coding environments thereby allowing for dealing with systems of any size while
retaining analytic nature. This is relevant for interpretable modeling,
analytic characterization and estimation, and engineering design in which the
objective is to tune parameter values to achieve specific behavior. Such
insights cannot directly be obtained using numerical simulations.
