Can a macbook run on mains with a dead battery?

Question

Iron

Actually I got mac 13 2009 late tyts
It has a practically dead battery, well, I'm used to that this is not a sentence, I plugged it into the network and everything works, but this one does not, it charges 12 hours to 100% then 20 minutes of work and does not even turn on.
Those who got the laptop say that it's the battery, but after googling a bit I found information about what the battery controller, power supply and the same battery can do.
I don’t really want to give the second cost of the laptop to the service now :)
What could it be?

Answer 1

if you have 2GB of RAM there, then the game is not worth the candle especially)

Answer 2

The point is not in the battery, but in the power controller.

Answer 3

Disassemble the beech and disconnect the battery from the motherboard, if that's the case. Works without battery too