Dell 505 will run on battery only and power cord is good

I'm afraid that the problem is with the Main board and that it must be serviced because if the adapter is able to charge the battery then obviously the system is detecting the charger but does not run on the charger alone indicates a faulty main board. May be you can try to update the BIOS and Chipset by going to the manufacturer's page and look for drivers and downloads for your model.

If this does not help then the main board either needs to be service or replaced.

I was just wondering why would you want to remove the battery out of the laptop when it is working fine. Let the battery be in the laptop and be charged by the adapter and use it without the power cable when you need it portable and connect it back when you need the power.

I would never remove a battery when it is working fine but if you are very much particular about running the laptop on power cable without battery then the problem is with the main board for which I advice you to contact the manufacturer for a replacement (if under warranty)

Try connecting the laptop to DC adapter and no battery inside. If it works , then you have a faulty battery, or a fault on motherboard power charger circuits.

If the laptop does not work with only DC adapter connected, that happens because there is a bad contact near to laptop DC plug, near to the point where the connector is soldered to motherboard.

In that case the laptop will not stay on when charger is connected, as the, or more commonly will not charge at all.

The only way to fix this kind of problem is

Laptops are not meant to be run without the battery installed. If your computer runs with the battery in and the AC adapter will charge the battery if it is low, there is no problem.

I hope I am not misreading your problem.