eigenfunction expansions for second-order boundary value problems with separated boundary conditions

In this paper, we investigate some properties of eigenvalues and eigenfunctions of boundary value problems with separated boundary conditions. also, we obtain formal series solutions for some partial differential equations associated with the second order differential equation, and study necessary and sufficient conditions for the negative and positive eigenvalues of the boundary value problem. finally, by the sequence of orthogonal eigenfunctions, we provide the eigenfunction expansions for twice continuously differentiable functions.