When you say "Y" class ticket, do you mean Economy Class or full fare Y?

Assuming you're talking about a regular discount Y ticket, you can't just turn in your original ticket and ask for a new one with new set of dates. You will need to buy up to the next fare class to do that provided that you still meet all the requirement of that new fare class. Otherwise, you can use the value of your old ticket towards the purchase of a new one (+ change fee if it's in the same fare class).