You sure can.
All you need is an Emulator, such as No$GBA, and the game's ROM, which you'll have to buy a card reader to get.
There may or may not be websites where you can download ROMs free, you'll have to look around.
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You can't play nintendo DS games on a PC; you can however find a modular for your PC, which will allow you to download games on the web to play on the PC.
You can not Pokemon Heartgold is a Nintendo DS game and you can only play PC games on the PC
Yes, the original Nintendo DS as well as the Nintendo DS lite can play NDSi games.
No. There is no way to download or play DSiWare games on Nintendo DS or DS Lite systems.
Yes, Pokemon SoulSilver is a Nintendo DS game so it can be played on a Nintendo DS