I am just very confused about this one. It is not a very difficult technique, but my answer disagrees with the text's answer section. I have the errata for the book (not many versions have been made yet), but it does not say there's a typo with this problem.
F(x) = e^(x) * sin(2x) + 3cos(2x)
My reasoning:
D(F) = 2e^x * cos(2x) - 6sin(2x)
D^2(F) = -4e^x * sin(2x) - 12cos(2x)
D^2(F) is also equal to -4F, and so the operation (D^2 + 4)F = 0 (right?).
So the annihilator A(D) = (D^2 + 4).
The answer section of the text says A(D) = (D^2 - 2D + 5)(D^2 + 4).
Is the text wrong? Or am I wrong somewhere? Thank you in advance!
F(x) = e^(x) * sin(2x) + 3cos(2x)
My reasoning:
D(F) = 2e^x * cos(2x) - 6sin(2x)
D^2(F) = -4e^x * sin(2x) - 12cos(2x)
D^2(F) is also equal to -4F, and so the operation (D^2 + 4)F = 0 (right?).
So the annihilator A(D) = (D^2 + 4).
The answer section of the text says A(D) = (D^2 - 2D + 5)(D^2 + 4).
Is the text wrong? Or am I wrong somewhere? Thank you in advance!