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For the following linear differential equation

$$a_ny^{(n)}+\cdots+a_1y'+a_0y=Q(x),$$

most books teach the method of undetermined coefficients, variation of parameters and Laplace transforms. Tenenbaum and Pollard's Ordinary Differential Equations teaches all these methods, but also the inverse operator method and has about 40 pages on it. I have never learned this method as an undergraduate but once I tried to teach it, I loved this method very much and it seems to me that in many cases, this method will be easier to use than other methods.

However, I do not find this method in many books on Differential Equations. Do people generally not teach this method in an undergraduate course? Why?

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    $\begingroup$ @user1027, thanks for the comment! Using the same logic, we can also omit the "undetermined coefficients" as what it can solve can be solved by other methods. $\endgroup$ – Zuriel Oct 23 '20 at 22:47
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    $\begingroup$ I think it is both interesting and useful for students to see this math. For those interested in quantum mechanics it has the added benefit of giving them experience working with operators. Where else would they see operators acting on an infinite dimensional space ? And yet, it's concrete. I don't think the topic has been reasoned out of DEqns, it is just part of the general plan to dumb down the course. Series will disappear for the same reason... not logic... simply retention. Of course, we can sell the loss of difficult techniques as a means to do more "modeling"... $\endgroup$ – James S. Cook Oct 24 '20 at 3:09
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    $\begingroup$ Fwiw, here is an example of how I have taught this stuff to an audience nearly devoid of math majors. I've taught DEqns this way to a few hundred engineers with good success. youtube.com/… $\endgroup$ – James S. Cook Oct 24 '20 at 3:17
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I'm just an ex-student, not a teacher.

But my hunch is that it's more user friendly after you know the other methods, rather than before them. Sort of like the "tic tac toe" method of integration by parts. It's good. But makes more sense, if you've had to slog through the detailed algebra several times before.

The issue then becomes one of time. One of the issues with a standard ODE course is just getting everything covered. If I had to pitch passsengers off the lifeboat, I would sacrifice operator method well before series solutions, Euler equation, etc.

P.s. Spiegel does cover it (I think, am out to dinner...but check when I get home). Edit: yes, he does. Calls it an alternate method, treats it after the others. Has several pages on it, but not the 40 page length of Tannenbaum.

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