posted by pakh .
u(x,y)=xln(x^2+y^2)-(2y times inverse of tan of y/x)
1)find derivative of u wrt to x and then wrt y
2)show that (x times derivative of u wrt x)+(y times derivative of u wrt y) = u +2x
u = x ln(x^2+y^2) - 2 y tan^-1(y/x)
du/dx = x(2x)/(x^2+y^2) +ln(x^2+y^2) -2 y(-y/x^2)(1/(y^2/x^2)) + 0 because dy=0 this time
That should get you started. I am not going to plod through the whole thing.