Solution techniques for f'(x)=f(g(x))

Question

I stumbled over this seemingly natural question and was surprised, that I couldn't find a satisfying answer. Differential equations of the type $f'(x)=g(f(x))$ are studied for all kind of classes of g, but what is there to say about solutions to the opposite case where $f'(x)=f(g(x))$? I even failed to find solutions for the very simple case $f'(x)=f(1-x)$, say on domain $[0,1]$. Ultimately I am looking for solutions to $ f'(x)=\frac{1-f(x)}{f(1-x)-1+x} $. Thank you for any helpful comments.