We have solved many initial-boundary value problems in which separation of variables has led to sines and/or cosines as eigenfunctions, in terms of which we had to write some given function (such as an initial temperature, position or velocity) in a Fourier expansion.

We will now solve initial-boundary value problems in which “new” functions, or special functions, have to be created as eigenfunctions. Further, we will have to expand some given function in a series of these new eigenfunctions. This section provides an environment in which we can deal with these issues.

Begin with a general form of the differential equation for the eigenfunctions. Separation of variables in many kinds of important problems leads to a differential equation of the form

where *r, q*, and *p* are given, continuous functions on the relevant interval, say [*a, b*] or (*a, b*), and *r*(*x*) > 0 and *p*(*x*) > 0 for *a* < *x* < *b*.

This is the *Sturm-Liouville differential equation*. Of course, any differential equation can be written in different ways, so we will refer to the form given in equation 6.1 as the *standard form* of the Sturm-Liouville equation.

A *Sturm-Liouville problem* consists of this differential equation, together with boundary conditions. We will consider two kinds of boundary conditions.

A *regular Sturm-Liouville ...*

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