# Definition:Sturm-Liouville Equation

## Definition

A classical **Sturm-Liouville equation** is a real second order ordinary linear differential equation of the form:

- $ (1): \quad \displaystyle - \frac {\mathrm d} {\mathrm d x} \left({p \left({x}\right) \frac {\mathrm d y} {\mathrm d x}}\right) + q \left({x}\right) y = \lambda w \left({x}\right) y$

where $y$ is a function of the free variable $x$.

The functions $p \left({x}\right)$, $q \left({x}\right)$ and $w \left({x}\right)$ are specified.

In the simplest cases they are continuous on the closed interval $\left[{a \,.\,.\, b}\right]$.

In addition:

- $(1a): \quad p \left({x}\right) > 0$ has a continuous derivative

- $(1b): \quad w \left({x}\right) > 0$

- $(1c): \quad y$ is typically required to satisfy some boundary conditions at $a$ and $b$.

### Weight Function

The function $w \left({x}\right)$, which is sometimes called $r \left({x}\right)$, is called the **weight function** or **density function**.

### Eigenvalues

The value of $\lambda$ is not specified in the equation.

Finding the values of $\lambda$ for which there exists a non-trivial solution of $(1)$ satisfying the boundary conditions is part of the problem called the Sturm-Liouville problem (S-L).

Such values of $\lambda$ when they exist are called the **eigenvalues** of the boundary value problem defined by $(1)$ and the prescribed set of boundary conditions.

The corresponding solutions (for such a $\lambda$) are the **eigenfunctions** of this problem.

## Source of Name

This entry was named for Jacques Charles François Sturm and Joseph Liouville.

## Sources

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Sturm-Liouville equation**