  The method to solve inhomogeneous linear differential equations that is
usually taught at school relies on the fact that the right hand side function
is the product of a polynomial and an exponential and that the linear spaces of
those functions are invariant under differential operators (finite or
ordinary).
  This short note uses Jordan's canonical decomposition to prove that the
linear spaces spanned by products of polynomial and exponentials are the only
linear complex spaces that are invariant under differential operators,
therefore non-homogeneous linear finite difference or ordinary differential
equations can only be generically solved when the right hand side belongs to
those spaces.
