PART ONE:

Step 1: Open a new Finder window by clicking the Finder icon in the dock.

Step 2: Select Go to Folder... from the Go menu.

Step 3: Input ~/Music/iTunes into the textfield then click the Go button.

Step 4: Double click the file named iTunes Music Library.xml to open it in TextEdit.

Step 5: Write down the value of the Library Persistent ID key. Take care not to modify the file before closing. It should look something like D501EB4887717F8F.

PART TWO (MAC):

Step 1: Open a new Finder window by clicking the Finder icon in the dock.

Step 2: Select Go to Folder... from the Go menu.

Step 3: Input ~/Music/iTunes into the textfield then click the Go button.

Step 4: First we will create a backup of iTunes Music Library.xml and iTunes Library. Select files and press Command+c then Command+v on your keyboard. This will make a backup copy of the files.

Step 5: Double click to open iTunes Music Library.xml.

Step 6: Make note of the current Library Persistent ID then change the Library Persistent ID key to the one we wrote down from Part One and save the file.

Step 7: Now we will need to open iTunes Library using a HexEditor. You can download HexEdit from here: HexEdit Download

Step 8: Once you have opened iTunes Library use HexEdit to search for the old Library Persistent ID we replaced in Step Five. Do this by clicking Find from the Menubar. Make sure to select Hex as your search type.

Step 9: Once found replace the old Library Persistent ID with the one from Part One and save the file.

Step 10: You can now launch iTunes and connect your iPhone to the new computer for syncing. Select your iPhone from the list of devices on the left. Notice if now select Manually manage music and videos from the Summary Tab you will no longer be prompted to erase your data, nor will the files on your iPhone be grayed out.