A second-order ordinary differential equation
![d/(dx)[p(x)(dy)/(dx)]+[lambdaw(x)-q(x)]y=0,](/images/equations/Sturm-LiouvilleEquation/NumberedEquation1.svg) |
where 
is a constant and 
is a known function called either the density or weighting
function. The solutions (with appropriate boundary conditions) of 
are called eigenvalues
and the corresponding 
eigenfunctions.
The solutions of this equation satisfy important mathematical properties under appropriate
boundary conditions (Arfken 1985).
There are many approaches to solving Sturm-Liouville problems in the Wolfram Language. Probably the most straightforward approach is to use variational (or Galerkin) methods. For example, VariationalBound in the Wolfram Language package VariationalMethods` and NVariationalBound give approximate eigenvalues and eigenfunctions.
Trott (2006, pp. 337-388) outlines the inverse Sturm-Liouville problem.
