I am travelling to US for ten days, and have to make application for?
Not unless your non-EU passport is Sudanese, Iraqi, Iranian, or Syrian, or you have been to those countries since 2011 (see United States Begins Implementation of Changes to the Visa Waiver Program). You can just use your EU passport to get into the US. The US does not legally recognize multiple citizenship, so by law, you enter the US using only one passport. You will be treated only as a national of the country you enter in on, using that passport. For US-EU dual citizens, by law, they have to enter only using their US passport for this reason, but they can enter Europe and leave the US on an EU passport.