As for "laptop can do in seconds" - well, not if your objective function takes a few minutes to run. The last time I needed it, the objective function took about 2 minutes, and there were 8 parameters we were optimizing over. Standard derivative based optimizing algorithms will require 9 invocations of the function per iteration. So one iteration of the algorithm took about 18 minutes. Certainly not "seconds". Of course, those evaluations could be done in parallel, so I just had them run on multiple cores, bringing it down to only a few minutes per iteration.
Without knowing something about the algorithms my library was using, I would have been totally lost (not to mention I would have likely picked the wrong algorithm for the job at hand).
But yes - I did not write my own algorithm - just used an off the shelf one. However, if you expect that someone who hasn't studied the topic can just use a random optimization algorithm and get things to work, you are mistaken.
Also, compared to that curriculum, the topics you mention have robust methods and stable libraries? So you can use what somebody else did. It is more likely that you need to know the gritty details if you solve PDEs, vs solving ODEs.