Apple MacBook "Core 2 Duo" 2.0 13" (White/06) Specs

Identifiers: Late 2006 - MA700LL/A - MacBook2,1 - A1181 - 2121

I bought this 2006 white macbook about a month ago. I ran it by the apple store and they ran their diagnostics on it and all checked out. I got it home and upgraded it form snow leopard to lion and upgraded the ram to 2GB from the 1GB it had came with. It gave me no issues during all of this. When I bought it - it did not come with a battery. So i ordered an apple macbook battery to stay OEM for the battery. That battery finally came 3 days ago. I had not booted or messed with the macbook in the meantime after initial set up. I put the battery in booted the macbook, and it ran great for about 3- 5 minutes. Then all of a sudden the screen went black, i checked to see if it just went dim but even shining a flashlight at the screen you could not see the desktop, but the indicator light on the front of it was solid - white so i knew it was on. So I powered it off and tried to boot again. This time the screen flickered for a second - i heard the optical drive sound and then nothing happended. I reset the ram and it then booted again and worked for about 3 - 5 min and the screen went black again - same thing all over again. Since then I have reset PRAM & SMC reset. Im not sure what could be wrong with it. Could it be the RAM ? I thought if RAM was bad it wont boot. Please let me know any throughts and possible solutions. I got this for my little girl to use for school and now need to put it into action for her. - the macbook is a 2006 late model 2GHZ with 2gb ram running Lion