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I am scheduled to teach a graduate-level course in engineering whose basis is in the solution of ODE’s and PDE’s, and thus is about halfway between a math course and an engineering course.

We introduce methods for solving these equations (perturbation expansions, self-similarity, series expansions, and so on) that they may not have had in taking ODE’s as an undergraduate.

Between that and the discrepancies in undergraduate curricula, are there guidelines to follow in figuring out how much material I should review from undergrad ODE’s in the graduate-level course?

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Even for math grad students, I'd forcefully review much more than many traditions seem to indicate. That is, I would not presume perfect recall of the standard curriculum, especially either in detail or in "big picture". Further, in my experience, even very smart people with unusually good memories greatly benefit from repetition. It's not "one and done", ... except for those occasional potentially misleading situations where one's thought processes had arrived at a juncture ripe for an epiphany.

Yet further, I've found that it is unfortunate to too aggressively assume that there's surely little need for review... after finding that people have supposedly assimilated various fancy things, but cannot do simple things (that are the background and/or foundation for the fancy things). I'm not advocating not mentioning fancier things, but, rather, advocating going back-and-forth between more sophisticated things and their more elementary antecedents (and examples).

In particular, I am no longer embarrassed about talking about as-simple-as-possible motivating/explanatory examples.

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Review basic ODEs and PDEs (Tenenbaum and Farlow, or the two Schaum's or Kreyszig.) Take a pre-look at Bender and Orszag text and at the Carl Bender videos on YT. That's enough. Umm...little bit of Taylor and McLaurin series might be wise if that has been a while.

P.s. This is a learning math question, not teaching math, but in before the close!

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    $\begingroup$ It’s definitely a “teaching” question (although for some reason the title made me think otherwise at first) $\endgroup$ – pjs36 Feb 12 '18 at 23:57
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    $\begingroup$ I'm the instructor, not a student! $\endgroup$ – aeismail Feb 13 '18 at 0:28
  • $\begingroup$ Just cover it as you go. "Remember Taylor series?" [shuffling of feet sound] "Well, a Taylor series is...And we use it in this problem..." You don't have time to reteach everything and it will frustrate the class if you hijack lectures to have review sessions. But if you make a few helpful comments like that, it is nice, since they don't remember it all. In course of course, they will get some re-practice in old stuff along with the perturbation methods, etc. $\endgroup$ – guest Feb 13 '18 at 1:28

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