As an engineer, I promise you, I give not one single solitary conscious fuck about the uniqueness of solutions to differential equations. Mostly I just hit things with Fourier or Laplace transforms as appropriate until they stop moving.
As an engineer working in some optimization problems where is utterly important if a solution is a local or a global minimum, I assure you I give many fucks about the uniqueness of solutions.
I totally agree with your way of working for most applications, it is what I do most of the time too, and I agree with the article in that this point is given more importance in DE courses than it really is for engineers, but there are perfectly valid use cases for these theorems in engineering.
I'm glad someone got something out of that section of the course besides a bad grade and the lingering suspicion that if it had been taught out of the engineering school it would have made more sense.
For ODEs you can simply think of solving the autonomous system numerically. Since I have many numerical algorithms to solve such a system, solutions exist. Since (most of) those algorithms are totally deterministic and offer no choices anywhere along the way (except maybe for some initial conditions) the solutions are unique.
For PDE's it's much more interesting, as the author points out.
I think the underlying issue is that the technical conditions guaranteeing for existence and uniqueness for ODEs (the Picard-Lindelof theorem) are so easy to satisfy (which is what guarantees that different numerical algorithms will give the same answer) that they're something most students are unlikely to encounter in practice.
That said, I do think there is some pedagogical value in teaching existence/uniqueness even though the result may not be so interesting because it shows students that it's possible to get information about solutions directly from the equation even without explicit formulas available. It also introduces them to the sort of abstract arguments at the core of modern mathematics.
There is definitely pedagogical value. This issue is covered by the author, but when studying ODEs for the first time, at some point the student will come across the fact that exp(x) is a solution to y' = y, which is easy enough, but needs to be convinced that exp(x) is the solution up to linear combinations. To most students this is not at all obvious! Lack of explanation here is doing the student a disservice.
