I have a vibration in my front differential that only vibrates when the engine is cold at a low speed. After 2nd gear, there is no vibration. I have changed the front differential 3 times but the vibration goes away for a few days and then it is the same as before. Any suggestions why is it happening.

 

What diagnostic work was done that indicates the front differential is bad? Unless they run out oil, and physically break, there isn't much to go wrong that would cause it to vibrate.

I'm assuming this a 250, what year, how many miles and what is the full history? A carboned up 250 with worn engine mounts is the most common cause of vibration. Clean the MAF sensor, get the intake and valves cleaned, replace spark plugs with new Denso and take a snapshot of fuel trims. That last item will tell more about engine health than anything.

After that, wheels, tires, brake system, half shafts, driveshafts, all have things that are far more likely to induce vibrations than the front diff.

gl