Foreword

Sometimes, it's easier to say how things change than to say how things are.

From 3Blue1Brown: Differential Equations

When you write down a differential equation, you're specifying constraints and information about e.g. how to model something in the world. This gives you a family of solutions, from which you can pick out any function you like, depending on details of the problem at hand.

Today, I finished the bulk of Logan's A First Course in Ordinary Differential Equations, which is easily the best ODE book I came across.

A First Course in Ordinary Differential Equations

As usual, I'll just talk about random cool things from the book.

Bee Movie

In the summer of 2018 at a MIRI-CHAI intern workshop, I witnessed a fascinating debate: what mathematical function represents the movie time elapsed in videos like The Entire Bee Movie but every time it says bee it speeds up by 15%? That is, what mapping $t \mapsto x (t)$ converts the viewer timestamp to the movie timestamp for this video?

I don't remember their conclusion, but it's simple enough to answer. Suppose $f (t)$ counts how many times a character has said the word "bee" by timestamp $t$ in the movie. Since the viewing speed itself increases exponentially with $f$ , we have $x^{'} (t) = {1.15}^{f (x (t))}$ . Furthermore, since the video starts at the beginning of the movie, we have the initial condition $x (0) = 0$ .

This problem cannot be cleanly solved analytically (because $f$ is discontinuous and obviously lacking a clean closed form), but is expressed by a beautiful and simple differential equation.

Gears-level models?

Differential equations help us explain and model phenomena, often giving us insight into causal factors: for a trivial example, a population might grow more quickly because that population is larger.

Equilibria and stability theory

This material gave me a great conceptual framework for thinking about stability. Here are some good handles:

Let's think about rocks and hills. Unstable equilibria have the rock rolling away forever lost, no matter how lightly the rock is nudged, while locally stable equilibria have some level of tolerance within which they'll settle back down. For a globally stable equilibrium, no matter how hard the perturbation, the rock comes rolling back down the parabola.

Resonance

A familiar example is a playground swing, which acts as a pendulum. Pushing a person in a swing in time with the natural interv

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