Find the annihilator given y''-6y'+10y = e^(3x)-x
My book list the annihilator as D^3(D-3) but I don't understand how they got that. I am wondering if it might be a typo in the text.
The Annihilator for $\displaystyle e^{3x}$ is $\displaystyle \left(D-3\right)$ and the annihilator for $\displaystyle -x$ is $\displaystyle D^{\color{red}2}$.
Now, the annihilator of $\displaystyle e^{3x}-x$ is the product of the two individual annihilators: $\displaystyle D^2\left(D-3\right)$
Note that $\displaystyle D^3\left(D-3\right)$ also works as an annihilator, but $\displaystyle D^3$ really isn't necessary.
Now if the question was find the annihilator for $\displaystyle e^{3x}-x^2$, then the answer your book gives would be correct.
Does this make sense?