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To describe the space of solutions to Af, one must know something about differential equations ; ;

that is, one must know something about D other than the fact that it is a linear operator.

However, one does not need to know very much.

It is very easy to establish by induction on R that if F is in Af then Af ; ;

that is, Af, etc..

Thus Af if and only if Af.

A function G such that Af, i.e., Af, must be a polynomial function of degree Af or less: Af.

Thus F satisfies Af if and only if F has the form Af.

Accordingly, the ' functions ' Af span the space of solutions of Af.

Since Af are linearly independent functions and the exponential function has no zeros, these R functions Af, form a basis for the space of solutions.