It could be because your 2DS or cartridge is damaged. Your game should work just fine. You should consider getting your cartridge checked out. But I'm sure your other cartridges work so it just must be your Pokémon White cartridge that is broken and not your 2DS (unless you're 2DS recently became broken and you don't know). Perhaps more information about your cartridge (and possibly your 2DS) may be helpful.
Source: Experience.
The Nintendo 2DS and New Nintendo 2DS XL play Nintendo 3DS games in 2D. They also play nearly all Nintendo DS and Nintendo DSi games. Compare Nintendo handhelds here.
Source
I put Pokémon Black version 2 into a plain 2DS and it loaded. 1. You can try cleaning the golden thingies on the back of your cartridge (make sure none are missing :P). 2. After that you should stick the cartridge in your 2DS and wait (about 1 minute or more if you want. Sometimes, it takes a little longer for a game to load). After 1 minute if it doesn't pop up take it out and repeat step 2. If it still doesn't work, then you need a new cartridge/2DS (if you were to get a new system, I recommend that you don't buy another 2DS).
I hope this helps a bit. Good luck!
