Piyush remembers that the birthday is after 17^th October but before 20^th October  , while Vishnu remembers that birthday is after 18^th October but before 22^th October . On which date of  October is the birthday?

19 October

According to question , 

According to Piyush birthday is after  17^th October but before 20^th October 

Possible date of birthday= 18 , 19      ------ ( 1 ) 

According to Vishnu birthday is after 18^th October but before 22^th October

Possible date of birthday = 19 , 20 , 21      ----- (2 ) 

from ( 1 ) & ( 2) 

Common satisfying date is 19 October

So birthday is on 19 October .