I just got to Figaro City & had redone the means to unlock Dragon Slave & I just had to post what I did...
OK, here it is, the complete step by step means of unlocking Dragon Slave:
1. Make sure you've already learned all Levels of Magic (Levels 1~10) by learning of them in the Magic Guilds & that one level from Liddick (level 4 or 5 I think).
2. Once in Figaro go first to the Hospital & talk to the old man in the upper right part of the hospital. You'll know it's him if he ends his dialog with "unless you're here to give me a password". This is the most important thing.
3. Now go to the Magic Guild & learn Level 10 magic (you could learn level 10 magic first before talking to the old man if you want as well) from the "wizard with personal issues" (the guy in the far left room at the Magic Guild's 3rd floor) as the Chancellor (the guy in the middle room) is rather busy.
4. Now head on down to the Magic Guild's basement library & go to the far right, talk to the guy walking around on the second row of shelves (while at it, learn Valmaceid in one of the books on the shelves as well). Talking to him will yield the "password" you'll need to give or rather, say to the old man in the hospital. The password by the way is, "Isn't that old man a pip?", too bad having read about it here on my post won't do you any good as you don't manually input it in game
5. Now go back to the old man in the hospital and chat with him again, this time around Lina will say the password & the old coot will have added new lines to his dialog. He'll reveal the location of "Dragon Slave", where to find it (Flagoon Tree of course) & who "created the spell" which is Ley Magnus of course
Congratulations! Now just play through the game until you can explore Flagoon Tree & go to the altar where Dragon Slave is at & you're done!
If someone ever updates the Slayers FAQ. this sort of detail should be included on finding the biggest spell in the game.