I think I might have the same problem. But my laptop only shuts down when it isn't plugged in(it's not instant either it usually goes off in about 15 min)...as long as it is plugged in it works fine. I bought a new battery about 7 months
Dell Inspiron 6000
Windows XP Service Pack 2
I had the same problem for over a month and could not find an answer on this forum so I went into the CPU forum of this site. Apparently the thermal compound between the CPU and the heatsink had dried out and no heat was being transfered. Once I cleaned both surfaces with alcohol and put an even coat of compound on both, the computer fired up and has been running nicely and faster for the last three weeks. I hope this helps.