System of equations where solution depends on derivatives and solution itself

Question

I wanne recover two functions $F_i : [0,1] \to \mathbb R$, $i = 1,2$ with $F_1(x) = F_2(1-x)$. The solution is given by \begin{align} F_1(x) = \phi_1(x,F_1(x),F_2(x),F'_1(x),F'_2(x))\\ F_2(x) = \phi_2(x,F_1(x),F_2(x),F'_1(x),F'_2(x)) \end{align} where $\phi_i(\cdot)$ is known, but too messy to put it here. What are these kind of problems and how would one solve for $F_i(\cdot)$? Addtionally the value $F_1(\frac{1}{2}) = F_2(\frac{1}{2})$ is given. May I could use a boundary value problem solver? It seems to be the case that this is some system of differential equations?!