i want to know if i can still buy the apple care, since my purchase date is 63 days already?

from: Buy an AppleCare plan - Apple Support

"In most countries and regions, you can buy AppleCare+ within 60 days of your device purchase. In Japan, you can buy AppleCare+ within 30 days of your device purchase. In some countries and regions, you can buy an AppleCare Protection Plan within 1 year of your purchase of an eligible Mac, Apple TV, or other device."

Note that Apple will likely want to make sure your device does not have damage before they sell you AppleCare. If it is damaged already it is too late.

Bob Timmons wrote:

"a little bit dead" is not as serious as "mostly dead". That's different from being "all dead" since "mostly dead is slightly alive".

Apologies to the Princess Bride

Referencing The Princess Bride is always a good thing in my book!

Lawrence Finch wrote:

Well, no, for 2 reasons: You are past the 60 days.

I left that part undecided since I don't see the bit where we know where the OP is located, and Apple does have that bit about a year in some places.

Well, no, for 2 reasons: You are past the 60 days, AND to add AppleCare+ the phone must be undamaged. Just like you can’t buy life insurance on someone who is dead.

Lawrence Finch wrote:

Just like you can’t buy life insurance on someone who is dead.

Even if they're only a little bit dead? ;-)

"a little bit dead" is not as serious as "mostly dead". That's different from being "all dead" since "mostly dead is slightly alive".

Apologies to the Princess Bride