It is shown that the eigenvalues and eigenfunctions for the class of “separable” or “semidegenerate” kernels can be determined from the solution of a linear differential equation, which is usually more amenable to machine solution. The theory is extended to solve a simultaneous diagonalization problem for two separable kernels. Finally, some new connections are obtained between Riccati differential equations and Fredholm integral equations of the second kind. The results specialize to previously known results for symmetric separable kernels.