Ive got a two year old latitude D630 that is shutting down right after the Dell splash screen. I googled this problem and saw someone suggested pressing down on the left side of the laptop while its booting up and that worked to the point where I got to the login screen, but after a few seconds, it just shut down again.
Try the dignostic boot: Press and hold the Fn key, press Power, release them both together. Should give you an idea what the issue is.
Sounds like you might a hard drive problem, as the left side is where its located. Try reseating the drive first if still no luck, if you can try another drive. The memory is located in the middle of the laptop and under the keyboard, so pushing the left side for a memory fault unlikely. Memory faults on D630's happen when their new or a few months old, after two years I think that's one you can rule out, Unless you've dropped it recently.
I don't know about the fn key but I tried F8 and when the opportunity came up to choose the last good boot up I selected it and the machine booted up properly and continues to boot up properly