# Definition:Bessel's Modified Equation

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## Equation

**Bessel's modified equation** is a second order ODE of the form:

- $x^2 \dfrac {\d^2 y} {\d x^2} + x \dfrac {\d y} {\d x} - \paren {x^2 + n^2} y = 0$

The parameter $n$ may be any arbitrary real or complex number.

## Solution

The solutions of **Bessel's modified equation** with parameter $n$ are known as **modified Bessel functions of order $n$**, and they are functions of the parameter $n$.

## Also known as

**Bessel's modified equation** is also referred to as **Bessel's modified differential equation**.

Some sources present it as **the modified Bessel equation**.

The parameter $n$ is variously presented. Some sources use $p$.

## Also see

- Results about
**Bessel's equation**can be found**here**.

## Source of Name

This entry was named for Friedrich Wilhelm Bessel.

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 24$: Bessel Functions: Bessel's Modified Differential Equation: $24.31$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**Bessel's equation** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Bessel's equation**