At long last, there is a new video. And around half an hour at that. This kicks off the series on differential equations, with a general overview of the topic, focused on a couple of specific examples from physics to love.
I remember how, as a student, most math classes I'd taken before ODEs focused on finding explicit solutions to problems, so my bias going in was that the primary goal to keep developing ever more intricate tools for finding such solutions. I think it took me a while to realize that even though there are many tactics for this, sometimes the most important problems actually can't be tackled with these methods, so arguably the better muscle to build is knowing how to reason about these equations without a solution.
So for this introductory video, I'm let that be the theme of things. In future videos, as we get into the main tactics for tackling these problems, I'd like to circle back to the idea of using the tools not just to find solutions, but also to study the problems which lack a clean, explicit answer. For example, studying linear equations is as much about understanding approximate local behavior of nonlinear dynamics as it is about linear systems themselves.