I would like to compute the eigenvalues and eigenfunctions for the Fredholm integral operator
$$K(g) = \int_0^1 e^{xt} g(t) \,dt.$$
The sources I've checked* seem to say that the process is fairly involved. Has anything been published on this kernel? Or, if not, am I correct that it's going to be a hard thing to do?
* See, e.g., equations (12) and on here: https://www.encyclopediaofmath.org/index.php/Fredholm_equation