We may apply further assumptions to our string vibration model to add realism and generality. In the simplest version, we assumed constant elasticity, no external forces (like gravity), and the like. It is possible to introduce such factors (p for density, q for external forces and r as another factor less easy to describe), and after similar calculations in the previous example, we end up with the following for the position equation:

-(p(x) y'(x))' + q(x) y(x) = k r(x) y(x)

where p, q and r are given functions of x, and we also assume that p and r are positive functions (meaning they only take values greater than zero). Note that the way the brackets are arranged, we take p times the derivative of y, then take the derivative of the result, which gives a second derivative term. If we now add the boundary conditions:

a y(0) + b y'(0) = 0
c y(1) + d y'(1) = 0

a, b, c and d are given constants, and k is the eigenparameter. This is a second-order differential equation on the interval from 0 to 1. It appears all over the place in physics, where the functions p, q and r represent certain physical quantities. This is called the regular Sturm-Liouville problem, and it has been a focus in terms of what I've been learning over the last year.

This problem was originally studied after it popped out of new heat transfer models developed in the 1840's, and has remained an object of research ever since. Those studying the problem are interested in what happens to the eigenvalues and solutions (or even if any exist) given different properties of the coefficient functions and by messing around with the boundary conditions.