Nonlocal Ordinary Differential Equations I would like to solve the following Ordinary (?) Nonlinear (?) Differential Equation:
$$\left[\int_0^1 u^2(\xi) \,d\xi\right]u_{xx}(x)+u(x)=1,$$
with $u(0)=0$ and $u(1)=1,$ where $u_{xx}$ denotes the second derivative of $u$ with respect to $x$. What is the procedure to solve it?
 A: Here's one method. Note that
$$\int_0^1u^2(\xi)\,d\xi $$
is independent of $x$. Let us replace it with $a,$ to give us a necessary degree of freedom. So, let's solve the IVP
$$a u''(x)+u(x)=1,\; u(0)=0,\; u(1)=1. $$
The solution is
$$u(x)=1-\cos \left(\frac{x}{\sqrt{a}}\right)+\cot \left(\frac{1}{\sqrt{a}}\right)
    \sin \left(\frac{x}{\sqrt{a}}\right). $$
Now then, let us see if we can force $\displaystyle\int_0^1u^2(\xi)\,d\xi=a.$ We have 
$$\int_0^1u^2(\xi)\,d\xi=\frac{1}{4} \left(-2 \cot ^2\left(\frac{1}{\sqrt{a}}\right)+6 \sqrt{a} \cot
    \left(\frac{1}{\sqrt{a}}\right)+4 \csc \left(\frac{1}{\sqrt{a}}\right)
    \left(\csc \left(\frac{1}{\sqrt{a}}\right)-2 \sqrt{a}\right)+2\right). $$
Plotting this up with $a\in[0,0.2]$ and the $y$ range being from $[0,2]$ shows us that there is a value of this that achieves $a,$ near $0.16.$ Using the following FindRoot command in Mathematica:
FindRoot[(2 + 6*Sqrt[a]*Cot[1/Sqrt[a]] - 2*Cot[1/Sqrt[a]]^2 +    4*Csc[1/Sqrt[a]]*(-2*Sqrt[a] + Csc[1/Sqrt[a]]))/4-a,{a,0.16}]

yields the result {a->0.175136}.
Naturally, this is not an exact solution, but you can make it as exact as you like.
