The function $u(x)$ is a solution of the second-order ODE
$$\alpha\frac{d^2u}{{dx}^2}+ru(1-u)-u = 0 $$
satisfying the boundary conditions $$\frac{du}{dx}\Bigg|_{x=0,L}=0$$
and is bounded in $0 < u(x) < 1$
I'm hoping to find a closed form for the average value of $u(x)$ over the domain $[0,L]$:
$$\bar u = \int_0^Lu(x)dx$$
I've reduced this to
$$\int_0^Lu(x)dx = \frac{r}{r-1}\int_0^Lu(x)^2dx$$
for some parameter $r$, and I'm unclear on where to go from here. I'd like to find a closed form for $\bar u$.
If this is intractable, is there a way to asymptotically expand this average value as $\dfrac{r}{r-1}\to\infty$?