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Does it stay powered on with only a curser or does it shut off then? If it shuts off, I'm going to venture, airflow/overheating issues. Has the laptop ever been disassembled and cleaned out? If not, after two years I would expect some dust bunnies residing within. A good cleaning may go a long way here. An easy way to check is, when you power on the unit, can you feel airflow from 3-4" from the exhaust vent. I did have a fan go out on a Vaio that was about two years old so that could've happened also.
I wouldn't think that the CMOS battery would be dead after only 2 years but replacing it may help (especially if you've aleady taken it apart to clean thoroughly).
There could easily be other issues but that is where I would start here.