Solution of ${\mathrm{y}}^2\mathrm{dx}+({\mathrm{x}}^2-\mathrm{xy}+{\mathrm{y}}^2)\mathrm{dy}=0$

$\begin{array}{l}(x^2-xy+y^2)dy=-y^{2}dx\\ \Rightarrow \frac{dx}{dy}=\frac{x^2-xy+y^2}{-y^2}\\ =-{\left(\frac{x}{y}\right)}^2+\left(\frac{x}{y}\right)-1\\ Putx=vy\\ \Rightarrow \frac{dx}{dy}=v+y.\frac{dv}{dy}\\ \Rightarrow v+y.\frac{dv}{dy}=-v^2+v-1\\ \Rightarrow \int \frac{1}{v^2+1}dv=-\int \frac{1}{y}dy\\ \Rightarrow Ta{n}^{-1}v=-\log y+c\\ \Rightarrow Ta{n}^{-1}\left(\frac{x}{y}\right)+\log y=c\end{array}$