This is of the form $Mdx+Ndy=0$

$$ M=x\sqrt{x^2+y^2}-y \\ N=y\sqrt{x^2+y^2}-x$$

$$\frac{\partial{M}}{\partial{y}}=\frac{x}{2\sqrt{x^2+y^2}}2y-1$$

$$\frac{\partial{M}}{\partial{x}}=\frac{y}{2\sqrt{x^2+y^2}}2x-1$$

$$\frac{\partial{M}}{\partial{y}}=\frac{\partial{N}}{\partial{x}}$$

The given Equation is an exact differential equation.

Solution is given by

$$ \int Mdx+\int(Terms\ in\ N\ not\ containing\ x)dy=c\\ y=constant \\ \int\left( x\sqrt{x^2+y^2}\right)dx+0=c$$

Put $x^2+y^2=t$

$\therefore 2xdx=dt$

$$ \int \sqrt{t}dt-(y)dx+0=c \\ \frac{1}{2}\frac{t^{\frac{3}{2}}}{\frac{3}{2}}-xy=c \\ \frac{\left( x^2+y^2\right)^\frac{3}{2}}{3}-xy=c$$