Over on the Reading Penrose blog, Jean Louis Van Belle (and I apologize if I’ve got the name capitalized or punctuated wrong but I couldn’t find the author’s name except in a run-together, uncapitalized form) is trying to understand Roger Penrose’s Road to Reality, about the various laws of physics as we understand them. In the entry for the 6th of December, “Ordinary Differential equations (II)”, he gets to the question “What’s Physics All About?” and comes to what I have to agree is the harsh fact: a lot of physics is about solving differential equations.
Some of them are ordinary differential equations, some of them are partial differential equations, but really, a lot of it is differential equations. Some of it is setting up models for differential equations. Here, though, he looks at a number of ordinary differential equations and how they can be classified. The post is a bit cryptic — he intends the blog to be his working notes while he reads a challenging book — but I think it’s still worth recommending as a quick tour through some of the most common, physics-relevant, kinds of ordinary differential equation.