N ways to solve a differential equation

Mark Dominus writes about how to solve the differential equation (f(x))² + (f^'(x))² = 1, which arises in the analysis of LC-circuits.
Go ahead, try it. There are at least four distinct solutions given there (bearing in mind the usual caveats about distinctness of solutions); I won't give any of them. If you want to know how I solved it, read Dominus' post, which ends with:

I would like to add a pithy and insightful conclusion to this article, but I've been working on it for more than a week now, and also it's almost lunch time, so I think I'll have to settle for observing that sometimes there are a lot of ways to solve math problems.

That's something definitely worth remembering.

As to why I ended up using a power series approach, which you'll see explained in the link: I'd been teaching the solution of differential equations by power series at the time, so it seemed natural to do that rather than to look for some kind of trick.