Windows Defender isn't a very deep or detailed antivirus. I would really suggest getting something else and you should consider malware detection software as well.
You may also wish to try starting in "Safe Mode" and see if it will stay up and running. Granted you can't run everything in "Safe Mode", but if it works fine and doesn't shut down in there, then you know it is something on the computer.
How to enter "Safe Mode" when booting the computer.
In Windows 8 and 10...
As your computer restarts, press F8 (possibly a few times) to enter "Safe Mode"
a. Press the "F4" key to Enable "Safe Mode".
(The computer will then start in "Safe Mode" with a minimal set of drivers and services.)
b. Press the "F5" key to Enable "Safe Mode" with Networking.
( Once "Safe Mode" with Networking starts, Windows is in Safe Mode, with additional network and services for accessing the Internet and other computers on your network.)
c. Press the "F6" key to Enable "Safe Mode" with Command Prompt.
(In "Safe Mode" with "Command Prompt" starts Windows in Safe Mode, with a Command Prompt window instead of the Windows interface. This option is mostly only used by IT professionals.)
Now sign in to the computer with your account name and password. (If you have one set.) When you are finished troubleshooting, you can exit "Safe Mode" restarting your computer.
In Windows 7/Vista/XP...
1. Immediately after turning on the computer, or restarting it (usually after you hear your computer beep), tap the F8 key, repeatedly, in 1 second intervals.
2. The computer will then display hardware information and run a memory test.
3. Next the "Advanced Boot Options" menu will appear.
4. In the "Advanced Boot Options" menu use the arrow keys to select "Safe Mode" or "Safe Mode with Networking" and press ENTER.