Computer won't recognize new iPhone in iTunes

I just received a new iPhone 6 in the mail to replace the damaged one I have. I followed the setup directions to get the new phone activated and erased and reset all of the content and settings on the old, damaged one. The problem is my computer won't recognize the new device in iTunes and my wife's computer will erase the content of it because it is synced to the library on my computer. How do I get my computer to recognize my new iPhone?