Question

Solve:

$(x^2+xy)dy=(x^2+y^2)dx.$

Answer 1

$\frac{dy}{dx}=\frac{x^2+y^2}{x^2+xy}$

$\frac{dy}{dx}$ is a homogeneous function

put $y=vx$

$\begin{array}{c}\frac{dy}{dx}=\frac{v+xdv}{dx}\\ \frac{v+xdv}{dx}=\frac{x^2+y^2}{x^2+xy}\\ \frac{1+v}{1-vdv}=\frac{1}{xdx}\\ -2\log \left|\left(\frac{y}{x}\right)-1\right|-\frac{y}{x}\\ \log \left|x\right|+c\end{array}$