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.