Just starting my study of Integral Equations on Rahman's book, I have studied how to find the resolvent kernel $R(x,t;\lambda)$ of the integral equation since its kernel $K(x,t)$.
The expression for the resolvent kernel is
$R(x,t;\lambda)=\sum_{n=0}^{\infty}\lambda^{n}K_{n+1}(x,t)$,
and
$K_{n+1}(x,t)=\int_{t}^{x}K(x,\tau)K_{n}(\tau,t)d\tau$.
The idea is to find $K_{n+1}(x,t)$ building up the recurrence ${K_{1}(x,t),K_{2}(x,t),...}$, where $K(x,t)=K_{1}(x,t)$. Once you have already found the expression for $K_{n+1}(x,t)$, you just have to substitute on the expression for $R(x,t;\lambda)$ and that's all.
I have tried to do this with $K(x,t)=\sinh(x-t)$ and $K(x,t)=e^{-(x-t)}\sin(x-t)$, but I'm stuck because I can't find the general expression for $K_{n}(x,t)$.
My question is how to go any further or if there's any other more suitable method for this cases, so maybe I don't have to build up any recurrence.
Thank you very much.