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.
