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Very nice! Thank you, David, for the lucid explanation.
I stand corrected.
I was certainly guilty of an error (ignorance), and to my mind the
worse error of being dogmatic in my ignorance. Well, well, I had
not seen this before, although I've used 3 different books in
teaching analytic mechanics and have referred to others many times,
I simply had missed it. As explained in my post, I was under the
impression that I had tried everything possible in attempting to
separate variables in this situation (not all shown here). But I
should not have been so quick to assume that!
The derivation of Kepler's Laws from Newton's Law of Universal
Gravitation, for instance, showcases some similar tricks and
manipulations to separate variables, and I give a 4-page handout on
that to my students, showing various false starts, some of the
logic behind the things we try.
It's an art form, not merely following predetermined processes.
Thanks, David, for adding to my appreciation of this particular
aspect of that beauty today. :-)
Ken