What causes central air conditioner duct to sweat only on one end of mobil home?

There could be an air leak to the outside air that is more humid. Try investigating the actual duct for any holes or cracks and use duct tape to seal them up and see if that helps the condensation problem you have.

If the A/C is running and the duct is cold it will attract the moisture in the air. Actually the moisture condenses on the duct surface since it is cold. Insulating the duct is one cure and if the duct is sweating under the trailer then a plactic 6 mil vapor barrier on the ground under the mobile home will be a huge help in more ways than one.