My MacBook isn't starting up!

I have a macbook, i think it's a 2009 model. Whenever it runs out of battery it won't start up again, every time i have to take it to the store to start it up again.

I was told the problem is not the battery but the mother board, and to change a motherboard will cost me just a littlbe bit less than a new macbook. Any tips, help and advices will be helpful thanks!!