When you "buy" a domain (actually register it), your registrar provides you with DNS service, typically on its own servers. But, you can relocate that to other DNS service provider or you can even setup your own DNS server on a VPS for instance.
The easiest solution in your case is probably to move your domain service from active-domain.com to a third party free DNS service provider which supports API based IP updates. There are many out there. Just make sure that your new DNS service provider accepts custom domain names (like the domain you got yourself) and provides an update client, which can be a simple script or an app, or a utility to run on a computer or router.
Alternatively, you can always move your registration to another registrar with such dynamic update capability. It would cost you an additional year's registration fee but you won't lose any money as it would extend your domain's lifespan.