The

*linear second order ordinary differential equation* of type

is called the

*Bessel equation*. The number

*v* is called
the

*order of the Bessel equation*.

The given differential equation is named after the German mathematician and astronomer

*Friedrich Wilhelm Bessel* who studied this equation in detail and showed (in 1824) that
its solutions are expressed through a special class of functions called

*cylinder functions* or

*Bessel functions*.

Concrete representation of the general solution depends on the number

*v*. Further we consider separately two cases:

- The order
*v* is non-integer;
- The order
*v* is an integer.

Case 1. The Order v is Non-Integer

Assuming that the number

*v* is non-integer and positive, the general solution of the Bessel equation can be written as

where

*C*_{1},

*C*_{2} are arbitrary constants and

*J*_{v}(*x*),

*J*_{−v}(*x*)
are

*Bessel functions of the first kind*.

The Bessel function can be represented by a series, the terms of which are expressed through the so-called

*Gamma function*:

The Gamma function is the generalization of the

*factorial function*
from integers to all real numbers. It has, in particular,

the following properties:
The Bessel functions of the negative order (−

*v*) (it's assumed that

*v* > 0) are written
in similar way:

The Bessel functions can be calculated in most mathematical software packages.
For example, the Bessel functions of the 1st kind of orders

*v* = 0 to

*v* = 4
are shown in Figure 1.
The corresponding functions are also available in MS Excel.

Case 2. The Order v is an Integer

If the order

*v* of the Bessel differential equation is an integer, the Bessel functions

*J*_{v}(*x*) and

*J*_{−v}(*x*)
can become dependent from each other. In this case the general solution is described by another formula:

where

*Y*_{v}(*x*) is the

*Bessel function of the second kind*. Sometimes this family of functions is also called

*Neumann functions* or

*Weber functions*.

The

*Bessel function of the second kind* *Y*_{v}(*x*)
can be expressed through the Bessel functions of the first kind

*J*_{v}(*x*) and

*J*_{−v}(*x*):

The graphs of the functions

*Y*_{v}(*x*) for several first orders

*v* are
shown above in Figure 2.

*Note*:
Actually the general solution of the differential equation expressed through Bessel functions of the first and second kind is valid for non-integer
orders as well.

Some Differential Equations Reducible to Bessel's Equation

**1.**
One of the well-known equations tied with the Bessel's differential equation is the

*modified Bessel's equation* that is obtained by replacing

*x* to −

*ix*.
This equation has the form:

The solution of this equation can be expressed through the so-called

*modified Bessel functions of the first and second kind*:

where

*I*_{v}(*x*) and

*K*_{v}(*x*)
are modified Bessel functions of the 1st and 2nd kind, respectively.

**2.** The

*Airy differential equation* known in astronomy and physics has the form:

It can be also reduced to the Bessel equation. Its solution is given by the Bessel functions of the fractional order

:

**3.** The differential equation of type

differs from the Bessel equation only by a factor

*a*^{2} before

*x*^{2} and has the general solution in the form:

**4.** The similar differential equation

is reduced to the Bessel equation

by using the substitution

Here the parameter

*n*^{2} denotes:

As a result, the general solution of the differential equation is given by

The special Bessel functions are widely used in solving problems of theoretical physics, for example in investigating

- wave propagation;
- heat conduction;
- vibrations of membranes

in the systems with cylindrical or spherical symmetry.