What is the preferred method of teaching "linear" differential equations today?

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?

• 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". Aug 22 '18 at 20:10
• 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. Aug 23 '18 at 3:54
• 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. Aug 28 '18 at 13:28