Does anybody know the solution of this singular Cauchy-like integral equation:
$$ y(x) = \int_{-\infty}^{\infty} \frac{y'(x')}{x-x'}dx'\\ y(0) = 1, \lim_{ \lvert x \rvert \to \infty } y(x)= 0 $$
The nominator of the integrand is the derivative of the function in the LHS. The function y(x) is symmetric, i.e., y(x) = y(-x). It monotonically decays from 1 to 0 like an exponent as x grows.
If the exact solution does not exist, any guess how to get a good approximation?
Motivation
I am solving a problem of fluid inflow to a straight slot in an infinite 2D medium. The equations become integral ones because of construction. I am deriving them considering first a point-source problem, and then integrating it over slot dimensions. As a result, I get two equations: for fluid pressure $p$ and inflow $u$: $$ \beta p''(x) = u(x)\\ p = \int_{-1}^{1} u(x') ln|x-x'| dx'\\ p(0) = 1, p'(\pm 1) = 0 $$
Here p(x) is symmetric, and u(x) is antisymmetric with respect to x=0.
In the limiting case of small $\beta$ (which means small conductivity of a slot), I turn to infinite boundaries, take integration by parts, and write the equation above for the pressure only, where y(x) = p(x).
My efforts
Meanwhile, I was managed to solve this equation only numerically. I want to know if the analytical solution exists.
Thank you very much.