- #1

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I'm studying from Prof. Arthur Mattuck's lectures, and it is the lecture #13 which is confusing me. He embarks on solving (if someone who has seen my other thread can he tell me whether I have used 'embark' (an intransitive verb) in a correct way?)

$$

\begin{align*}

y^{''} + Ay^{'} + By = e^{\alpha x} &&\textrm{where α is a complex number}

\end{align*}

$$

And writes the LHS as ##P(D) y## which I understand totally. The case when ##P(\alpha) \neq 0## is clear to me, the solution is

$$

y_p = \frac{e^{\alpha x} }{P(\alpha)}

$$

But when he moves to the case when ##P(\alpha) = 0##, I get a little confused. Though, I can prove

$$

\begin{align*}

P(D) e^{ax} u(x) = e^{ax} P(D+a) u(x) && \textrm{ it's no longer α, we got a there, but it is still complex}

\end{align*}

$$

Now, I don't understand things from time : 37:00 in the linked video. My doubts are thus:

- He says "if a is simple root of P(D)" then the solution is ##y_p = \frac{x e^{ax} }{P'(a)}##. How can ##a## be a root of ##P(D)## which is an operator? The operator have a null space not a solution set. We should say if ##a## is a solution of polynomial ##P(x)##.
- The second doubt is concerning the differentiation of ##P(D)##. I still don't understand what does a derivative of an operator mean, and above that with respect to another operator ##D##.