If the computer posts and makes no error beeps, it is probably shutting down to protect the processor. If it beeps check the manual's beep code page.
I have serviced computers that shut down soon after booting by removing a layer of caked on dust from the heat sink. If the problem is dust, remove the fan and clean off the outer sections of the heat sink. A thin piece of rigid plastic wrapped in a tissue works well to clean in between the fins once the top layer is removed. Clean the fan carefully, making sure no dust remains on the fins. Clean the case fans and blow the dust out of the box. Don't forget the chipset and video card while you are there.
Consider making dust filters for the case fans. A cheap and easy way is to get a mesh filter for a fish tank and cut to fit the fan. It helps to keep the inside of my case clean.
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