The solution x^2dy/dx=x^2+xy+y^2 is:

$\frac{dy}{dx}=1+\frac{y}{x}+(\frac{y}{x})^2$ Substitute $\frac{y}{x}=t⇒\frac{dy}{dx}=t+x\frac{dt}{dx}$

$⇒x.\frac{dt}{dx}=1+t^2⇒\int\frac{dt}{1+t^2}=\int\frac{dx}{x}$

$⇒\tan^{-1}t=ln\,x+c⇒\tan^{-1}\frac{y}{x}=ln\,x+c$