In my research (in differential geometry) I recently came across the following nonlinear second order ode: $$\frac{f''(x)}{f'(x)}-\frac{2}{x}+\frac{f'(x)+1}{2f(x)-x-1}+\frac{f'(x)-1}{2f(x)+x}=0$$ It actually arose from the symmetry reduction of some pde. I know from an analysis of the equation that there exists a 1-parameter family of solutions. Moreover I also know two explicit solutions; $$f(x)=x+\frac{1}{2}$$ $$f(x)=\frac{1}{4}+\frac{1}{4}(1+3x)\sqrt{(1+2x)}$$ The existence of these 2 solutions, expressible in terms of elementary functions, makes me wonder if one can in fact find more (if not all) explicit solutions to this ode. Note that both these solutions are well defined at $x=0$, although the ode itself is singular at that point! It is not too hard to show that any solution well-defined at $x=0$ requires $f(0)=\frac{1}{2}$ and $f'(0)=1$.

As far as I am aware there are no standard tricks for these type of fully nonlinear odes. I have been trying to simplify the ode by various substitutions but without any success.

I was hoping that someone might be able to spot a clever transformation, or even argue that it is impossible to find any other explicit solutions. I would also be interested to know of any references where such a class of ode that might have been studied.

This also leads me to ask if there is any general theory known about when can a solution to an ode (say of second order) be expressed in terms of elementary functions, or is it just a case-by-case study? Thanks!