how can one solve the integral
$ f(x,y)+K(x,y)+\int_{0}^{x}K(x,t)f(t,y)dt =0$ (1)
so $ q(x)= 2\frac{d}{dx}K(x,x) $ (2)
$ -y''(x)+q(x)y(x)=0 $ (3)
$ y(0)=0=y(\infty) $
$ q(x) $ here is the pontential of a Sturm Liouville operator (3)
$ \phi (x) = \int_{-\infty}^{\infty}\frac{d\lambda}{\lambda}(1-cos(\sqrt{x}t)\rho (\lambda)$ (4)
$ f(x,y)= \frac{ \partial _{x}^{2}\phi (x+y)+\partial _{x}^{2}\phi (x-y)}{2}$
here i have a doubt, inside the Gelfand-Levitan equation what is $ \rho (x) $
also is there an asymptotic or analytic solution to this equation ?? thanks.