Say I have a nonhomogeneous ODE:(adsbygoogle = window.adsbygoogle || []).push({});

y^(n) + ... + a1 y' + a0y = x

Define the differential operator Dx = x',

and p(D) = D^n + a_n-1 D^n-1 +...+a1 D + a0 I

Let C be the set of functions that can be differentiated as many times as we want.

Given Lemma: D - c I : C-->C is onto for c in complex.

Prove for all x in C there exists a solution y in C.

Is the following correct?

Write p(D) = (D - c1 I)(D - c2 I)...(D - cn I)

By the lemma, p(D) is onto.

If p(D) is onto, then there exists a y in C such that p(D)x = y.

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# Linear Algebra and DE

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