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Linear Algebra and DE

  1. Apr 24, 2005 #1
    Say I have a nonhomogeneous ODE:
    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.
  2. jcsd
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