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Are differential equations considered calculus and included in a calculus class or is it its own class? Also, if it is its own class then what calculus classes does it come after?

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In the US, it has become common to introduce differential equations within the first year of calculus. Usually, there is also an "Introduction to Ordinary Differential Equations" course at the sophomore level that students take after a year of calculus. The details of this can vary a lot between institutions.

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What part of differential equations is included in calculus may simply reflect the need of engineering or other majors at your institution. For example, when I taught Calculus II at NCSU in Raleigh, North Carolina USA, we covered how to solve: $$ ay''+by'+cy = 0 $$ where $a,b,c$ are constants. If you look in the major calculus texts, you'll likely find an appendix on constant coefficient second order ODEs. Certainly it is within the grasp of students in Calculus II. To be totally honest, it's way easier than a lot of the integration and integral-calculus-based problem solving which is typical of Calculus II. In short, if you can solve a quadratic equation and follow a recipe based on those solutions then you can just write down the general solution $y = c_1y_1+c_2y_2$. Frankly, you could put the method in highschool algebra books if you really wanted to, the method involves pretty much zero calculus (modulo it's derivation).

When facing the corresponding non-homogeneous problem: $$ ay''+by'+cy = g $$ where $a,b,c$ are constants and $g$ is the forcing function, the solution has the form $y = c_1y_1+c_2y_2+y_p$. We cover two methods to find $y_p$ (the particular solution), the method of undetermined coefficients and variation of parameters. Undetermined coefficients is pretty much just educated guessing with a little differentiation and a lot of algebra. Variation of parameters, in-contrast, is a general method which solves all inhomogeneous problems up to an integral you may or may not be able to hack in practice.

Before all this, we also covered a little about visualizing differential equations with direction fields, how to solve separable first order ODEs and the integrating factor method for solving linear first order problems.

In short, these are the major calculational tools that solve probably 90 percent of the basic applied problems you'll run across in basic engineering. A mass on a spring with friction, or an RLC-circuit, both are described by constant coefficient second order ODEs, so naturally engineering wants their students to know this math before the course so they can go deeper into the intuition and analysis of such problems. The first order ODEs are also applicable to a vast bank of examples across a wide variety of majors.

I would also point out, in Calculus III we typically discuss conservative vector fields as $\vec{F}$ for which there exists $f$ with $$ \vec{F} = \nabla f = \left\langle \frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z} \right \rangle $$ For example, if $\vec{F} = \langle x,2,z^2 \rangle$ then we need to find $f$ for which $$ \frac{\partial f}{\partial x} = x, \qquad \frac{\partial f}{\partial y} = 2, \qquad \frac{\partial f}{\partial z} = z^2 $$ This is a system of partial differential equations. It is standard to learn how to solve this in Calculus III. In this sense, we can't isolate the study of PDEs from the natural discussion of vector calculus. Btw, the solution is easily seen by guessing to be $f = \frac{1}{2}x^2+2\frac{1}{3}z^3+c$ where you can choose $c$ however you like. Furthermore, there are calculus texts which cover other ODEs later in Calculus III. For example, the exact equation $Mdx+Ndy=0$ is nice to analyze around the same time that Green's Theorem is discussed.

Anyway, last paragraph aside, it's not the math major that drives DEqns into calculus, it's these applications.

Incidentally, if we are to envision Calculus for math majors, I think the answer would be radically different than the adhoc collection of inertially propelled topics which forms our current canon.

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  • $\begingroup$ I didn't realize you had been at NCSU. I was there Fall 1989 to Summer 1993 as a graduate student (my 4th graduate program, and 3rd and finally successfully completed Ph.D. program). I was there at the tail end of a group of people who were interested in classical point set theory and general topology topics; all the NCSU dissertation titles I've seen since the mid 1990s, although I haven't looked in recent years, have been in far-distant and mostly applied areas. $\endgroup$ – Dave L Renfro Nov 18 at 7:27
  • $\begingroup$ @DaveLRenfro cool, I did not realize you went to NCSU as well. I was there undergrad from 1998-2001 then again for math grad school 2003-2008. I think you're right in assessing the applied drift. However, maybe they're trying to build back up the pure again from some hires I've seen lately. This much I know, it's not too easy to get accepted for my students who don't do great on the subject GRE... $\endgroup$ – James S. Cook Nov 18 at 19:11
  • $\begingroup$ The subject GRE might be treated more strongly now than it used to. None of the handful of better students I knew, who planned to attend graduate school when I was an undergraduate (this being around 1980) studied for it or worried about it (or the regular GRE). It seemed the main thing was how much beyond the minimal undergraduate stuff did you did for letters of recommendation (especially good was being the top student in more than one graduate class), but it sure seems something a lot of people study hard for now, at least given what I read in math SE. I took it "cold", for what it's worth. $\endgroup$ – Dave L Renfro Nov 18 at 21:25
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Both.

Differential equations are defined in the second semester of calculus as a generalization of antidifferentiation and strategies for addressing the simplest types are addressed there. But you do a more indepth analysis in a separate course that usually is called something like Introduction to Ordinary Differential Equations (ODE). That course is independent from the typical third semester of calculus in three dimensions and you could take the two courses in parallel. (There are also a separate category called partial differential equations, which has its own series of courses that will wait for after all of this.)

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    $\begingroup$ Largely independent from multivariable calculus, but partial derivatives can turn up in the context of exact differential equations. $\endgroup$ – J W Nov 6 at 12:31
  • $\begingroup$ Also, you might want to clarify the difference between systems of ODEs and single PDEs regarding "multiple dimensions". $\endgroup$ – J W Nov 6 at 12:36
  • $\begingroup$ Thanks! This really helped! $\endgroup$ – Luke Justin Nov 6 at 12:44
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    $\begingroup$ @JW Yeah, I focused so much on the discrete side of pure math that I never actually took a class on PDEs. I made my comments about that much more general so as to not be wrong. $\endgroup$ – Matthew Daly Nov 6 at 13:17

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