Can you provide us with more details? Right now I'm thinking you have the setting in power settings set so that the monitor will turn off after a certain amount of time. The secondary monitor working supports this. So like dark said, check your power settings or provide us with more details if this is not the problem.
If you have checked your power savings mode and it is set correctly (each OS is slightly different but similar) and it still happens even when its plugged in, then most likely your LCD power inverter is failing. If it was your back light, that generally just goes out completely and won't come off and on. Either way not to expensive or difficult to replace. The inverter on the most Dell laptops is soldered on to the LCD panel as a ground. Removing it is the hardest part of the whole thing.
And yes I am a Dell Certified Tech, just giving some general free advise.
After checking you power saver options. Take a good look at your screen, can you seen an image that is not illuminated by the backlighting? If you can the invertor or the lamp may be failling. Try to move the screen its hinges to see if the you get the screen to work,if you can the ribbon cable that sends the signal from the motherboard may be at fault. Replacing the invertor and lamp can be problematic, since it is easy to damage the LCD screen in attempting a replacement of the lamp and invertor.
It's highly possible your LCD screen has crapped out. I find LCD screens on laptops to be quite fragile (no matter how "tough" they are supposed to be made). Have you tried moving the laptop screen to a position that it might stay on? sometimes moving it back and forth will get it to stay on - if it works that way - then most likely you may want to send it back to get repaired.