  Real-world phenomena can often be conveniently described by dynamical systems
(that is, ODE systems in the state-space form). However, if one observes the
state of the system only partially, the observed quantities (outputs) and the
inputs of the system can typically be related by more complicated
differential-algebraic equations (DAEs). Therefore, a natural question
(referred to as the realizability problem) is: given a differential-algebraic
equation (say, fitted from data), does it come from a partially observed
dynamical system? A special case in which the functions involved in the
dynamical system are rational is of particular interest. For a single
differential-algebraic equation in a single output variable, Forsman has shown
that it is realizable by a rational dynamical system if and only if the
corresponding hypersurface is unirational, and he turned this into an algorithm
in the first-order case.
  In this paper, we study a more general case of single-input-single-output
equations. We show that if a realization by a rational dynamical system exists,
the system can be taken to have the dimension equal to the order of the DAE. We
provide a complete algorithm for first-order DAEs. We also show that the same
approach can be used for higher-order DAEs using several examples from the
literature.
