Answer
False
Work Step by Step
Let us take an example:
$A_1(D)F(x)=(D-1)(e^x+1)=-1$ and $A_2(D)F(x)=D(e^x+1)=e^x$
We can see that $A_1(D)$ and $A_2(D)$ do not annihilate $F(x)$.
Therefore, the given statement is $\bf{False}$.
Differential Equations and Linear Algebra (4th Edition)
by Goode, Stephen W.; Annin, Scott A.
Published by
Pearson
ISBN 10:
0-32196-467-5
ISBN 13:
978-0-32196-467-0
Chapter 8 - Linear Differential Equations of Order n - 8.3 The Method of Undetermined Coefficients: Annihilators - True-False Review - Page 525: e
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