$$ \frac{dy}{dx}=\frac{y(y-2x)}{x(x-2y)} $$
$$ \frac{dy}{dx}=\frac{y^2-2xy}{x^2-2xy} $$
Substitution und deren Ableitung.
$$ v= \frac yx$$
$$ y= vx$$
$$ \frac{dy}{dx} = x \cdot \frac{dv}{dx} + v $$
Einsetzen:
$$ \frac{dy}{dx}=\frac{y^2-2xy}{x^2-2xy} $$
$$ x \cdot \frac{dv}{dx} + v =\frac{y^2-2xy}{x^2-2xy} $$
$$ x \cdot \frac{dv}{dx} + v =\frac{( vx)^2-2x( vx)}{x^2-2x( vx)} $$
$$ x \cdot \frac{dv}{dx} + v =\frac{ v^2 \, x^2-2x^2 v}{x^2-2x^2 v} $$
$$ x \cdot \frac{dv}{dx} + v =\frac{ x^2(v^2 \, -2v)}{x^2(1-2 v)} $$
$$ x \cdot \frac{dv}{dx} + v =\frac{v^2 \, -2v}{1-2 v} $$
$$ x \cdot \frac{dv}{dx} =\frac{v^2 \, -2v}{1-2 v} -v $$
$$ x \cdot \frac{dv}{dx} =\frac{v^2 \, -2v}{1-2 v} -v \frac{1-2 v}{1-2 v}$$
$$ x \cdot \frac{dv}{dx} =\frac{v^2 \, -2v}{1-2 v} -\frac{v-2 v^2}{1-2 v}$$
$$ x \cdot \frac{dv}{dx} =\frac{v^2 \, -2v}{1-2 v} +\frac{2 v^2-v}{1-2 v}$$
$$ x \cdot \frac{dv}{dx} =\frac{3v^2 \, -3v}{1-2 v} $$
$$ \frac{1-2 v}{3v^2 \, -3v} \, \frac{dv}{dx} =\frac{1}{x} $$
$$ \frac{1-2 v}{v^2 \, -v} \, \frac{dv}{dx} =\frac{3}{x} $$
Integration nach dx auf beiden Seiten:
$$\int \, \frac{1-2 v}{v^2 \, -v} \, \frac{dv}{dx} \,dx =\int \, \frac{3}{x} \,dx $$
$$\int \, \frac{1-2 v}{v^2 \, -v} \, \,dv =\int \, \frac{3}{x} \,dx $$
