[jsiva]

You can model passive circuits with RLC ordinary differential equations (depending on the setup of the equations there's some algorithms such as Runge Kutta to solve them numerically). Afaik you can model some active components with ordinary differential equations, but I wouldn't be surprised if you had to resort to partial differential equations (More or less a "complete" electromagnetic simulation at that point).

[BSEdlMMldESB]

you're onto something

I was looking at some of the open documentation[1] and they show schematics and differential equations alongside each other.

I just keep burning out whenever I try to make sense of these things;

[1] https://the-analog-thing.org/THAT_First_Steps.pdf