I am giving my actual problem, but I would also like to understand how to solve such problems in general. I am neither a mathematician nor do I know a lot about Mathematica.
I am looking for a function $y(x,n,q)$ and I have the following equation solving $\overline x$ $$ \frac{2}{3} = \sum_{k=0}^{q-1} {n \choose k} (1-\overline x)^k (\overline x)^{n-k}, $$ and $y(x,n,q)$ should be equal to $$\frac{1}{1- \sum_{k=0}^{q-1} {n \choose k} (1-\overline x+ \int_x^{\overline x}\frac{3-y(z,n,q)}{3} dz )^k (\overline x- \int_x^{\overline x}\frac{3-y(z,n,q)}{3} dz )^{n-k}},$$ and combined these two then yield $y(\overline x,n,q)=3$.
My approach was to convert this into the following differential equation with $\frac{\partial u(x,n,q)}{\partial x}=y(x,n,q)$, i.e, using
$$ \int_x^{\overline x}\frac{3-y(z,n,q)}{3} dz = \overline{x} - x -\frac{u(\overline x,n,q)-u(x,n,q)}{3}, $$
such that
$$ \frac{\partial u(x,n,q)}{\partial x} = \frac{1}{1- \sum_{k=0}^{q-1} {n \choose k} (1- x - \frac{u(\overline{x},n,q)-u(x,n,q)}{3} )^k (x + \frac{u(\overline{x},n,q)-u(x,n,q)}{3})^{n-k}}. $$
Let us say I want to solve this for $q=2$ and $n=10$.
NSolve[{2/3 == Sum[Binomial[10, k] (1 - xbar)^k (xbar)^(10 - k), {k, 0, 2 - 1}], 1 >= wbar >= 0}, xbar, Reals] gives me 0.8821589036184635` and
NDSolve[{y'[x] == 1/(1 - Sum[ Binomial[10, k] (1 - x - (y[0.8821589036184635`] - y[x])/3)^ k (x + (y[0.8821589036184635`] - y[x])/3)^(10 - k), {k, 0, 2 - 1}]), y'[0.8821589036184635`] == 3}, y, {x, 0, 1}] // Simplify NDSolve::icordinit: The initial values for all the dependent variables are not explicitly specified. NDSolve will attempt to find consistent initial conditions for all the variables. NDSolve::idelay: Initial history needs to be specified for all variables for delay-differential equations.
while the same with DSolve gives me
DSolve::bvnul: For some branches of the general solution, the given boundary conditions lead to an empty solution.
However, at least in my imagination, the function I am looking for should not be that complicated. So I am at a loss why Mathematica has an issue here. What am I doing wrong?
EDIT: The curve I am looking for should intuitively look like this 
This is how it would look like in MatLab, but I would love to play with it in Mathematica. q=3; n=10;
syms C k; summand = nchoosek(n, k)*(1-C)^k * C^(n-k); C_sum = sum(subs(summand, k, 0:q-1))-2/3; assume(C, 'real'); assume(C > 0); C_sol = solve(C_sum, C); C = C_sol syms L x; summand_nom = nchoosek(n, k)* (1-C+int((3-L)/3, x, C))^k * (C-int((3-L)/3, x, C))^(n-k); nom = 1 - sum(subs(summand_nom, k, 0:q-1)); L = 1/nom; figure() eval_x = linspace(0,1,100); plot(eval_x, subs(L, x, eval_x)) 
