eckiller
Apr24-05, 04:59 PM
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.
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.