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By solving Partial Differential Equations by Separation of Variables, one reduces the PDE in several ODEs. Very common ODEs that originate from PDEs are special functions like

• Bessel's Differential Equation
• Legendre's Differential Equation
• Laguerre Differential Equation
• Chebyshev Differential Equation

This ODEs that they belong to a greater class of Oridinary Differential Equations, the so called Sturm-Liouville Differential Equations, thus the discussion of special functions will lead us to the Sturm-Liouville theory.

Let us consider a general ODE of the form \begin{align} y''+r(x)y'+q(x)y+ \lambda \rho(x) y=0 \tag{1} \end{align} In analogy to the Integrating Factor method to solve ODEs, we multiply (1) by the Integrating Factor $u=e^{\int r(x) dx}$. \begin{align} u(x)y''+u(x) r(x)y'+u(x) q(x)y+ u(x)\lambda \rho(x) y=0 \end{align} Then we can write the same equation as \begin{align} (u(x)y')'+\underbrace{u(x) q(x)}_{\tilde{q}(x)}y+ \lambda \underbrace{u(x)\rho(x)}_{\tilde{\rho}(x)} y=0 \end{align} with $(u(x)y')'=u(x)y''+u(x) r(x)y'$ since $u'=r \, u$

This is the Sturm-Liouville equation \begin{align} \left [ u(x)\frac{dy}{dx} \right]'+\tilde{q}(x)y+ \lambda \tilde{\rho}(x) y=0\\ \end{align}

Sturm-Liouville equations have the special property, that their eigenfunctions are orthogonal. This key-property allows to expand arbitrary functions from initial conditions into an orthogonal series. In the following a few well known equations are represented in Sturm-Liouville form

• Simple harmonic equation

\begin{align} (y')'+\omega^2y=0\\ \end{align}

• Legendre's equation

\begin{align} \left [ (1-x^2)\frac{dy}{dx} \right]'+\ell(\ell+1) y=0\\ \end{align}

• Bessel's equation

\begin{align} \left [ \xi\frac{dy}{dx} \right]'+ \left( a^2 \xi - \frac{n^2}{\xi} \right) y=0\\ \end{align}