When you see some infinite nested pattern, you try to capture its symmetries in some way. Differential equations are one such way. Or looking for symmetries. Induction can also help, or looking for fixed points. They can all give information. It’s a bag of tools. One could go and then look at the differential equation in more detail and relate it geometrically to the series and maybe learn some cool things (but that would be a posteriori, which you don’t want!)
Also, the differential equation doesn’t come from nowhere/is not random, it’s derived in the presentation from differentiating the function and seeing that it still resembles the original function in some way, allowing you to describe it with a differential equation.
What he describes as a trick with solving the differential equation can be explained - if you have f’(x)=A(x)f(x)+b, that’s a strong hint there’s an exponential there somewhere; if A is x, then the chain rule hints that you have and x^2 in the exponential, etc...
A lot of it (depending on what mathematician) can boil down to pattern matching and having a big-enough bag of tricks.
Also, the differential equation doesn’t come from nowhere/is not random, it’s derived in the presentation from differentiating the function and seeing that it still resembles the original function in some way, allowing you to describe it with a differential equation.
What he describes as a trick with solving the differential equation can be explained - if you have f’(x)=A(x)f(x)+b, that’s a strong hint there’s an exponential there somewhere; if A is x, then the chain rule hints that you have and x^2 in the exponential, etc...
A lot of it (depending on what mathematician) can boil down to pattern matching and having a big-enough bag of tricks.