Based on your initial query, I think its of value to note that there are two categories of problem regarding white screens on macbooks.
I've done some extensive research into getting my particular macbook air a1237 to register something on the primary display to no avail. In gist, it basically boots perfectly into the mac os (sounds and all.. which I'm able to see and work with on an external display) yet the screen itself remains white.. with some faint lines. I've hence found that the
two problematic "white screen" issues are:
a) The white screen of death
b) The white screen despite a fully functional mac (as with external display)
I will attempt in this response, to list some helpful fixes for both these problems.
a) The white screen of death. This problem is pretty common in the macbooks, macbook pros and macbook airs. Its catalyst lies in the computer's inability to boot of the hdd due to some form of corrupt data. Hence plugging into an external monitor does not solve anything. The solution would be an attempt to get the mac to boot off the harddrive in the event that its boot priorities have been screwed up (resulting in the white screen of death). The solution is understood better in the mac's inability to boot primarily off of any desired medium through default unless it is specified within the boot options.
Step 1: verify the harddrive inside the mac is fully functional by pulling it out and plugging it into another machine to see if it registers. If not, get a new one.
Step 2: I got this off another forum. This is pretty much the quote: When harddrive is removed,"Pleae correct me if I'm wrong, but, as far as I've guessed and investigated, it works like this: your Macbook Air's firmware (probably also any other Macbook, but I can't be sure about that) is always instructed to try to, in the first place, boot off certain piece of hardware: be it the hard disk, an external DVD unit, the install USB stick, the network, or whatever. This option can be set in the OS's Startup Manager. When the actual boot setting is any other than the hard drive, then, at boot time, the firmware will look for the instructed medium to boot off, and if for whatever reason it doesn't find it, then it will try to boot off the hard drive as a "last resource". BUT when the actual boot setting is "hard drive" and, for whatever reason, the data is corrupted or something is wrong with the boot sector, then the computer will NOT try to find any alternative device, or a network, where to boot off: it will keep forever trying to boot off the hard drive, and thus you get the dreaded white screen of death. The Apple engineers are so smart!!
But it's actually not dead: neither your motherboard nor your drive are dead. They're just in deep coma.
How can you wake it up from this coma? Simple: remove the back cover and disconnect the hard drive. Believe me: with the right screwdriver, it's very easy to do and even you can do it. Then, and only then, if there is NO hard drive where to try to boot off, the firmware is programmed to look for some other medium: USB stick, external DVD, network... So, plug this medium in and... voilá! There you get your little spinning animated icon and, eventually, the Apple's apple; the startup thisk will boot and there you are anew to life.
Now, once you've managed to boot, the FIRST thing you have to do is go to the Startup Manager and set the firmware to boot off your whatever startup medium is (otherwise, upon reboot you'll be in the same S*ITuation). After this, you can plug back in your hard drive and reboot your computer. Hopefully it will boot off as instructed (the USB/DVD/Network), it will recognize your hard disk, and you can use the Disk Utility to repair it, reformat it, or whatever you want to do with it. After this, everything will go back to normal".
b) White screen fix, yet working with external monitor.
The main problem here is in the feed that travels directly from the motherboard to the screen. You would likely find that this problem occurs if your screen was once broken, and you decided to replace it. Popped in the replacement and it was white with faint lines. Brightness can be controlled though.
Step 1: disassemble to the point where you are able to access all connects that run from the motherboard to the screen. The LVDS cable that runs from the logic board (mobo) to the screen should first be checked for any damages or looseness from the sockets on each of its ends. If this reseating or replacing the cable does not solve the problem, consider step 2.
Step 2: The problem may be in the port to which the cable is connected on the logic board. That little vga port/socket is sometimes really annoyingly infamous for its troubles. The fix would be an attempt to reflow its solder joints, so that it maintains its original functionality. I honestly have no idea how to actually reflow that joint/port/socket with a heat gun or oven, so I'd opt for Step 3. The basic idea though, is that by reaffirming the solder connection on the logic board, the vga signal would correct itself. This is what some users have reported.
Step 3: Logic board replacement sounds big and dreary, but in the end, I'm of the opinion that the time and convenience saved through buying a replacement logic board is more worth all the sacrifices and failure reflowing would bring. Since the lvds/vga port on the logic board is pretty sensitive, reflow may ruin the vga output to primary screen altogether. A replacement board on ebay would cost between 150-350 dollars. I don't mind getting refurbs or used ones provided they've been tested to 100% functionality.
Hope this helps