From one my idol and team mate here!!!
Common problem for iPhone 3Gs shorted mainboard it's beacause of PMic spoiled due to some 3rd party accessories being used such as car charger, emergency charger or any device that can spoiled the PMic (Power Management IC).
Normally if the PM ic is shorted, it getting hot when you charge or plug in to usb.
iPhone becomes fast low batt, but if you charge the battery and put back it powers on.
Just change PM ic...
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