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

Characterization of finite dimensional subspaces of complex functions that are invariant under linear differential operators