Commentary for "Justifying the Method of Undetermined Coefficients"

[MarkFL]

This topic is for questions and comments pertaining to this tutorial:

http://mathhelpboards.com/math-notes-49/justifying-method-undetermined-coefficients-4839.html

 

[Ackbach]

Good stuff! Just a couple of minor edits I'd recommend:

1. In the table, in the $y_{p}(x)$ column for Type I's, an $x^{s}$ seems to have become an $x^{2}$.

2. I would rewrite Equation (3) as follows (you haven't really used operator notation, but have included the test function in your definition of $L$, which is not usual):
$$(3) \quad L[y] \equiv a_nD^{n}+ a_{n-1}D^{n-1}+ \cdots+ a_{1}D+ a_0.$$
You do this later on, so this is more of a consistency thing, I think.

 

[MarkFL]

Thank you! For some reason, I want to enter a 2 instead of an s. I appreciate you catching this!

Your suggestion of being consistent with operator notation is an excellent one.

I have made both edits.

 

[Ackbach]

Hehe. Actually, I'm not sure I was consistent just then! You could either write
$$L \equiv a_{n}D^{n}+a_{n-1}D^{n-1}+ \dots + a_{1}D + a_{0},$$
or
$$L[y] = \left( a_{n}D^{n}+a_{n-1}D^{n-1}+ \dots + a_{1}D + a_{0} \right)y.$$

 

[MarkFL]

I prefer the first notation. This was a group project taken from my old ODE textbook, and the original notation came from there (notice how I am "passing the buck?").

I truly appreciate your suggestions, and feel the post has been improved because of them.