I have encountered the following equation, which resembles an ODE:
$$\int_0^1 (F^{−1}(x)−x)~ dx \cdot \frac{1}{F^\prime(F^{−1}(y))}=−C \frac{F^{\prime \prime}(y)}{(F^\prime(y))^{3/2}}$$
Here $C$ is some constant and $F$ is defined on the interval $[0,1]$. I don't expect to be able to solve this in general, but under the boundary conditions $F(0)=0$ and $F(1)=1$, then $F(y)=y$ is a solution, regardless of $C$.
My question is whether this solution is unique given these boundary conditions? I was under the impression that the standard uniqueness theorems of ODEs do not cover this sort of thing, so I was wondering if there is a generalization that does consider equations of this form.