Sturm-Liouville Eigenvalues via a Phase Function

One of the most common methods for calculating the eigenvalues of the Sturm-Liouville problem such as ( pψ ' )^ + ( λ r - q )ψ = 0 is to integrate the differential equation with trial values for λ. The appropriate boundary condition having been imposed at one end, the merit of the trial value for λ is judged by how nearly the corresponding function ψ fits the boundary condition at the other end. The present article describes an alternative procedure based on the differential equation satisfied by a phase function θ where tan θ = ψ/pψ '.