This is my wife's Asus Eee PC notebook. It recently began shutting down spontaneously. An Internet search suggests others have also had this problem. Often the suggested fix has been simply to get a new computer. But, in our case, the problem turned out to be a frayed cord inside the strain relief, which is a very inexpensive problem to fix.
My wife noticed her computer seemed to cut out when she happened to bump the power cord. There is a blue LED on the power converter brick that indicates it is working. While she was showing me how moving the power cord often caused the computer to shut down I happened to notice the blue LED went out at the same time. Wiggling the strain relief on the load side of the power converter frequently caused the blue LED to light again.
Screwdriver for prying
Third hand to hold wires for soldering
Step 1: Crack the Case
I decided to crack the case on the converter brick. I used a hacksaw to cut along the seam. Then I used a screwdriver to break the rest of the case seam.