The spectral theory for the Schrödinger equation on the half-line is treated through an analysis of the asymptotics of quadratic forms in a pair of solutions. Solutions of the second- and third-order differential equations for these forms are derived. In the case of the second-order DE (Milne's equation), it is shown that a single solution leads to the determination of the singular spectrum; this generalizes previous results which applied only to isolated points of the discrete spectrum. For the absolutely continuous spectrum, it is shown that a single solution allows one not only to locate the spectrum, but also to determine the spectral density function explicitly.