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

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.