I'm wondering if the following integral equation has any hope of an algebraic solution:
$\frac{2}{(x-2)^2}=\int_0^{\frac{1}{2}} f(x-s) f(s) \, ds$, where $f(\cdot)$ is unknown.
This is a Fredholm-type equation but with an unknown kernel. The helpful bit is that the kernel is equal to the unknown function...
The solution to this is a way to de-convolute the p.d.f. of two i.i.d. random variables x and y, when the pdf of x+y is known. Very useful...!
Any help is greatly appreciated, Pedro