DSolve[-(x*(x^2 + y[x]^2)) - y[x]*Sqrt[1 + x^2 + y[x]^2] + (-(y[x]*(x^2 + y[x]^2)) + x*Sqrt[1 + x^2 + y[x]^2])*Derivative[1][y][x] == 0,y[x],x]