It is possible to plug in a 3.5mm power adapter, but the main power connector is a different size - so in theory using an original PSU you shouldn't be able to do that. My guess is you have one of those multi connector adjustable voltage PSUs?
What happens typically is the power destroys the cap on the sound output, and sometimes the speaker (as had happened to a faulty unit I repaired previously). You really need to open it up and see if you can work out where the burning smell is coming from. It's not a good sign that it doesn't power up. Maybe a cap is shorted or something.