After a successful install, your agent logs, located at <db_agent_home>/logs, should contain this message:
Started AppDynamics Database Agent Successfully
If the agent log file is not present, the Database Agent may not be accessing the Database Agent command properties. To troubleshoot, check the application server log file where STDOUT is logged. It will have the fallback log messages, useful for troubleshooting the agent.
Verify that the Agent is Reporting to the Controller