iMac G5 PowerPC

When it turns itself off, and you power it on after, open Console (/Applications/Utilities).
Go back in time just before it powered off.
What shows as the last lines?
Look for 'shutdown' in the logs, does it list a line with the cause number?
 
Pop the back off and look for blown capacitors. If there are no blown/bulging caps it is probably just a dying power supply and those run about $100.
 
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