TITLE: A weird differential equation with some unknown trick?
QUESTION [5 upvotes]: Here is the differential equation I need to solve:
$$y=xy'+k\cdot \frac{y'}{y'-1}$$
(Here $k$ is a given constant).
What I have thought initially is to rewrite $\frac{y'}{y'-1}$ as $\left(1+\frac{1}{y'-1}\right)$. But I don't think it helps that much. Hope someone could give me a hint.
Also, in the question, we are asked to consider $k>0$ and $k<0$ separately. There is definitely a reason why behind this, but I have not figured that out.
Thanks :))

REPLY [5 votes]: $$y=xy'+k\cdot \frac{y'}{y'-1}$$
Is of the form
$$y=xp+f(p)$$ where $p=y'$
This is Clairaut's differential equation.