Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2012 / Paper 1 / 10 / i / Solution

Write the energy form as $Q[\psi ]={\int }_{\alpha }^{\beta }(p|{\psi }^{\prime }{|}^2+q|\psi {|}^2)dx$ and the weighted squared norm as $N[\psi ]={\int }_{\alpha }^{\beta }w|\psi {|}^{2}dx$. For real coefficients and appropriate homogeneous boundary conditions, the Sturm-Liouville problem is the stationary-point equation for their Rayleigh quotient $Q/N$.