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Proving linearity of the differential operator $a_n(x)D^n+a_{n-1}(x)D^{n-1}+\ldots+a_1(x)D+a_0(x)$.

I am taking a first course on ordinary differential equations(with background in linear algebra). I would like to know, if the proof to the below claim is correct (and that I am not using advanced facts to prove basic facts). Prove that every expression of the form: $$a_n(x)D^n+a_{n-1}(x)D^{n-1}+\ldots+a_1(x)D+a_0(x)$$ defines a linear transformation from $C^n[a,b]$ to […]