x^2+y^2=2(x+y)+xy =>
2x^2+2y^2=4(x+y)+2xy =>
(x^2-2xy+y^2)+(x^2-4x+4)+(y^2-4y+4)=8
(x-y)^2+(x-2)^2+(y-2)^2=8
(x-2)^2 y无解
x=2 => y=4
x=3 => y无解
x=4 => y=2
故x^2+y^2=2(x+y)+xy 的正整数解为(2,4)或(4,2)