please help me solve the differential equation I'm having a slight issue trying to solve this diffrential equation. I would greatly appreciate any help or tips to solve this problem.
$$\frac{x-f(x)}{(1-x-f(x))} = f'(x)$$

This is nearly a homogeneous DE. Set $f(x)-\frac12=(x-\frac12)u(x)$, then you should get a separable DE.
$$
(x-\tfrac12)u'(x)+u(x)=\frac{u(x)-1}{u(x)+1}.
$$
In general you can insert arbitrary shifts in $x$ and $f(x)$ and solve the linear system that reduces the sum of the constants to zero, so that the resulting equation becomes properly homogeneous (in the sense of a fraction of equal-degree expressions)
