Homogeneous type equations

Question

Solve the differential equation $\frac{dy}{dx}+y/x=y^2/x$ with $y=2$ and $x=1$.

Question should be solved using homogeneous type equations, probably choosing $y=V(x)\cdot x$. My problem is that i do not know what to do with $y^2/x$.

Answer 1

$$\frac{dy}{dx}+\frac{y}{x}=\frac{y^2}{x},$$ or equivalently $$\frac{dy}{dx}=\frac{y^2-y}{x},$$ use separation of variables now $$\frac{dy}{y(y-1)}=\frac{dx}{x},$$ that is $$\ln|y-1|-\ln|y|=\ln|x|+c.$$ Impose the initial data now to conclude $$-\ln(2)=c.$$ Thus $$\ln|y-1|-\ln|y|=\ln|x|-\ln|2|.$$

Answer 2

write your equation in the form $$\frac{dy}{y^2-y}=\frac{dx}{x}$$ Can you finish?