> he just happened to know that repeatedly using the differential operator on a seemingly random differential equation would create a pattern that resembles the original equation after using substitution repeatedly
Yep. One of the main differences of advanced Math is that the problem gives no clues about the solution. You look at the problem, you pick one of the standard tricks from your backpack of trick, and try to hit the problem as hard as you can. Sometimes the trick solves the problem. Sometimes the trick simplifies the problem. (Sometimes it is not obvious that it is a simplification). Sometimes the trick does nothing, so you just pick another trick from your backpack of tricks...
If that doesn't work, you call a friend that has another backpack full of tricks ...
The idea is that in a Math BS or PhD you see a lot of tricks, and get some advice about where each one can be useful. Transforming an infinite sum to a function is an standard trick. Transforming that to a differential equation is not so standard, but I've seen it before.
And sometimes no trick solves the problem, so you must invent a new trick. After a few year, if the trick is useful in other problems it will become popular and it will be added to the standard curriculum of a major in Math, or to the advanced classes for PhD, or just be a standard trick in a small niche.
There are two ways to solve a math problem and you described one of them: hit the nut with a hammer as hard as you can until it breaks. Grothendieck preferred the second method of slowly raising the ocean and soaking the nut for a few years, until one day it opens all by itself.
Homeopathic approaches can feel just as arbitrary, though. “How did the mathematician know to take a bazillion tiny steps to get here?”, “Which steps in this process are the ones that do anything?”, and “Why can’t they just get to the point?!” are all questions one might ask oneself while in the midst of reading such material.
Yep. One of the main differences of advanced Math is that the problem gives no clues about the solution. You look at the problem, you pick one of the standard tricks from your backpack of trick, and try to hit the problem as hard as you can. Sometimes the trick solves the problem. Sometimes the trick simplifies the problem. (Sometimes it is not obvious that it is a simplification). Sometimes the trick does nothing, so you just pick another trick from your backpack of tricks...
If that doesn't work, you call a friend that has another backpack full of tricks ...
The idea is that in a Math BS or PhD you see a lot of tricks, and get some advice about where each one can be useful. Transforming an infinite sum to a function is an standard trick. Transforming that to a differential equation is not so standard, but I've seen it before.
And sometimes no trick solves the problem, so you must invent a new trick. After a few year, if the trick is useful in other problems it will become popular and it will be added to the standard curriculum of a major in Math, or to the advanced classes for PhD, or just be a standard trick in a small niche.