I'm trying to solve the following problem:
Find the resolvent kernel, and solve the integral equation:
\begin{equation} u(t) = p(t) + \int_{0}^{x}a(t)b(s)u(s) ds. \end{equation}
Here we have an integral equation with a separable, Hilbert-Schmidt kernel with a self-adjoint integral operator. I know the standard way of solving integral equations and/or finding eigenfunctions of self-adjoint intgeral operators is to convert the equation into a differential equation. However, I'm not sure how to go about solving this question with this approach. The functions $a(t)$ and $b(s)$ aren't specified, and I'm not sure how this approach would work.
Any suggestions? Please only provide hints as this is a homework problem.