It depends on what you consider your "DB", is it the database application or the content.
The latter is easy, the content needs to be persisted outside the lifetime of the application. The convention used to be to have a "data" container, which simplified linking it with the application (e.g. using the Docker Engine create command --volumes-from parameter). With Docker 1.9 there is a new volume API which has superceded the concept of "data" containers. But you should never store your data in the overlay filesystem (if not only for persistence, but for performance).
If you are referring to a database application, you really enter a semi-religious debate with the microservices crowd. Docker is built to run single process. It is built for 12-factor apps. It is built for microservices. It is definitely possible to run more than one process in a container, but with it you have to consider the additional complexity of managing/monitoring these processes (e.g. using an init process like supervisord), dealing with logging, etc.
I've delivered both. If you are managing the container deployment (e.g. you are hosting the app), it is actually less work to use multiple containers. This allows you to use Docker's abstraction layers for networking and persistent storage. It also provides maximum portability as you scale the application (perhaps you may consider using convoy or flocker volume drivers or an overlay network for hosting containers across multiple servers). If you are developing a product for distribution, it is more convenient to deliver a single Docker Repository (with one Image). This minimizes the support costs as you guide customers through deployment.