The funny thing about Differential Equations is that pretty much everyone who teaches it has a different conception of what it covers.

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I mean, sure, there’s a common thread up through 2nd-order homogeneous ODEs via characteristic polynomial, but beyond that, DiffEq is like a box of chocolates and you never know what you’re gonna get.

I had always assumed that most people taking DiffEq courses were learning it fairly comprehensively: integrating factors, undetermined coefficients, variation of parameters, oscillators, Cauchy-Euler, Bernoulli, systems, Runge-Kutta, power series solutions, Laplace transforms, Fourier series, eigenfunctions, Lotka-Volterra (predatory-prey model), phase portraits, etc.

But when I did a bunch of tutoring, and talked to people taking DiffEq at various universities, what would typically happen instead is: after covering 2nd-order homogeneous ODEs via characteristic polynomial, the course would shoot off into whatever direction was most applicable to the instructor’s research interest.

And it wouldn’t cover all the other foundational stuff that you would reasonably be expected to know if you took a solid DiffEq course.

Hell, when I took DiffEq myself in college, it was taught as an afterthought by a hardcore algebraist who framed it like this: “Let’s learn about this algebraic structure called metric spaces. By the way, differential equations is a thing you can do in metric spaces. Linear independence, Wronskian, yadda yadda. Memorize these proofs. Okay, moving on!”

No modeling. None. Nada. No phase portraits. No Laplace transforms. No Fourier series. Not even frickin’ undetermined coefficients.

Anyway, it feels nice to finally put out a DiffEq course that’s comprehensive and sets the record straight on what it means to learn DiffEq.

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