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For the purposes of this question, there are two kinds of differential equations: linear, and non-linear, which is to say that there could be two ways to teach the subject.

One way is to separate linear from non-linear differential equations, and combine the former with other courses; e.g. by adding simple linear differential equations to the end of a calculus course or by combining their study with linear algebra. A more advanced course would then cover topics such as series solutions, or phase plane methods, laPlace transforms, etc.

The other way is to teach the fairly simple solution methods for (many) linear differential equations in the same course as the more complicated topics. That's the way I was taught (decades ago. But it seemed incongruent to me to put the simpler methods for solving linear equations in the same course as using the more advanced methods to solve non-linear equations,even though they fell under the same heading.

Is differential equations still (mostly) taught this way today, or have many college programs reorganized the teaching of differential equations along the lines suggested in the second paragraph instead?

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    $\begingroup$ When I was at NCSU as a graduate student circa 2003-2008 we were expected to cover 2nd order linear constant coefficients both homogeneous and nonhomogeneous in Calculus II. Engineering wanted it covered fairly early. I would guess the course you took is still fairly popular, my evidence would be textbooks. There are doubtless exceptions, for instance Blanchard/Devaney/Hall's text seems to be missing certain calculational techniques because they want to focus more on "models". $\endgroup$ – James S. Cook Aug 22 '18 at 20:10
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    $\begingroup$ Most books I know still teach diffyQs the traditional way. I think this is most helpful to science and engineering students. Even for math students, I doubt the en passant opinions within your question. It is very common for mathies here to assume that once they already understand something it should be taught in a more (seemingly) logical but less pedagically progressive manner. Because they are more attuned to math elegance than psychology of learning. $\endgroup$ – guest Aug 23 '18 at 3:54
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    $\begingroup$ Laplace transforms and series solutions are usually used to study linear ODEs, in a first course on ODEs, so I am not sure I completely understand the question. The traditional curriculum of the standard ODE course is dictated principally by the needs of mechanical and electrical engineering students. Forced, viscously damped spring systems and RLC circuits can be modeled by inhomogeneous second order linear ODEs, and for this reason a thorough understanding of such equations is basic for engineering students and the curriculum reflects this. $\endgroup$ – Dan Fox Aug 28 '18 at 13:28
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In the US, the norm has been (and still is) for there to be an INTRO to ODEs within the second semester of a stereotypical calculus course. You can see this as long ago as the Granville text from over 100 years ago. And it was almost identical in the Thomas Finney text I used in the 1980s, which was also AP aligned. (AP calc now still requires an ODE intro, but has backed off on second order constant coefficient homo/nonhomo requirement, versus what was required back in the day.)

I believe the rationale for this was that some students (then and still) only have two semesters of calculus. And there is the capacity to allow a short intro to ODEs (and to series/sequences). This gives the kids who never go further (bio and econ and chem majors and such), something.

For the physics and engineering and math (and at the Academies, all the kids), they will get a calc 3 and ODE course. The ODE course DOES have overlap with the intro from second semester calc. However, such overlap is not the end of the world. HS chem and bio and physics overlap with college (or AP) chem, bio, and physics. For most of the kids, stuff is hard enough. Seeing some parts of it twice is a feature, not a bug. (We are closer to dogs than to silicon...we are meat.)

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  • $\begingroup$ A lot of the bio or econ folks are likely to take the one or two semester sequence of calculus with little to no trigonometry and as I recall the "calculus for life science and or business" course (Tan for example) does not really do much anything with DEqns... of course it ought to, everything is DEqn's and we have robust tools to understand these numerically/visually... $\endgroup$ – James S. Cook Feb 10 at 5:38

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