Perturbation theory for the Sturm-Liouville problem with variable coefficients (Q1842355)

Using the example of the problem \((d^ 2/dx^ 2+ \lambda r(x)) \psi(x)= 0\), \(\alpha_ a \psi(a)- \psi'(a)= 0\), \(\alpha_ b \psi(b)+ \psi'(b)= 0\), \(r> 0\), with coefficient \(r(x)\) of fairly arbitrary form, we consider the possibility of an analytic solution of the Sturm-Liouville problem by the methods of ordinary perturbation theory.