Yes, testing with a known good PSU would be an excellent first step. Diagnosis is pretty much a process of elimination, so eliminating the possibility of a PSU fault is a good first approach.
The usual next step if the substitute PSU makes no difference would be to disconnect peripheral components to see if the system will start up each time and complain about the absence of an operating system. By disconnecting the drives and if it has a video card, removing that and connecting to the mainboard video, that will determine if it can boot up reliably, and adding a keyboard should then let you access the BIOS. If it happily does this every time, you could refit in turn video card, each drive until it played up.
If it still has problems just booting up the mainboard and RAM, we know the problem lies there somewhere. Bad caps would be a candidate then, even if none looked bad. The symptoms are not very indicative of a RAM fault, but the memory could also be removed to see if it starts up and shows an error for the missing RAM, usually beeping. If it still kept shutting off without RAM fitted, it would be looking like probably a mainboard fault.http://downloads.dell.com/manuals/all-products/esuprt_desktop/esuprt_inspiron_desktop/inspiron-531_owner%27s%20manual_en-us.pdfhttp://www.dell.com/support/home/ed/en/edbsdt1/product-support/product/inspiron-531/research
Edited by Platypus, 16 September 2017 - 07:43 AM.