How do you get the sick neglected dragon on dragon cave

You get one by killing an egg or hatchling and having it survive. It can be any egg.
(The method above is for Undead Dragons, with which Neglected Dragons are often confused. The method used to get Undead Dragons barely works except on Halloween. Below is the method for Neglected Dragons.)

No one is completely sure, but it seems that the way you get one is to get an egg and instantly fog it, and then log out and don't log back on until the day it's going to die and has a few hours or so left. Then wait for it to get down to 0 days, and put it in several emergency rooms of dragon cave fan sites. Then log off again and wait. When you log back on after a while, it MIGHT have turned neglected, or it might have died after not turning neglected at all, or it might the egg might have turned neglected and died, or it might have hatched as a Neglected and then died. But if you're really lucky, it both turned neglected and lived to adulthood. So, in a nutshell, to get a Neglected Dragon, you need effort, determination and pure Dragon Cave addiction. The main problem is that most eggs that "turn" (become Neglected) do so in their last minute of life, so it is hard to get them to live. The Neglected Science project has been experimenting methods to get this rare breed of dragon.