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?