No you can not solve the quest with "outside knowledge" ... this is a broken quest for me.
I found the first potion and gave it to one cow. Without a Scholar/Mystic character I missed the recipe to make the second potion. Later after killing the witch I wondered what to do with the witchs eye in my inventory and found the recipe as "outside knowledge". I crafted then the second potion. When talking to the other cow, I have the dialog option for giving it to her and she says that it takes time for the potion to see an effect. But the potion is still in my inventory and the quest can not be progressed.
Okay, I'll amend the statement: you CAN solve it if
you make the second potion before talking to the cows. I verified that this works.