Surprised nobody has mentioned this yet, but if you use (or can move to) Namecheap for your domain registration, they provide free DNS service that includes dynamic DNS [1]. The script to set it is merely an HTTP GET to a URL. A lot of routers provide that ability as part of their dynamic DNS set up (I know dd-wrt does, as does the Mikrotik router I currently own).

[1] https://www.namecheap.com/support/knowledgebase/category.asp...