I have an increasing function $f$ between $0$ and $1$. I am looking for the general solution of the following differential equation:
$$ \frac{x}{1-x} \frac{1-f(x)}{f(x)} = \frac{ f'(\frac{1+ x}{2})}{f'(\frac{ x}{2}) } $$
I show that the family of quadratic functions $f_a(x)=(2-a) x+(a-1)x^{2}$ solves this differential equation for all $a$ between $0$ and $2$. I would like to show that $f_a$ is the general solution of this equation.