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   Bessel Differential Equation
The linear second order ordinary differential equation of type
Bessel's differential equation
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
General solution of the Bessel equation for non-integer order
where C1, C2 are arbitrary constants and Jv(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:
Series solution of the Bessel's equation
The Gamma function is the generalization of the factorial function from integers to all real numbers. It has, in particular, the following properties:
a property of the Gamma function
The Bessel functions of the negative order (−v) (it's assumed that v > 0) are written in similar way:
Series solution of the Bessel's equation for negative non-integer order
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.
Bessel functions of the first kind of order 0 to 4
Bessel functions of the second kind of order 0 to 4
Fig.1
Fig.2
Case 2. The Order v is an Integer
If the order v of the Bessel differential equation is an integer, the Bessel functions Jv(x) and J−v(x) can become dependent from each other. In this case the general solution is described by another formula:
General solution of the Bessel's equation for any real order
where Yv(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 Yv(x) can be expressed through the Bessel functions of the first kind Jv(x) and J−v(x):
Representation of the Bessel's functions of the 2nd kind through the Bessel's functions of the first kind
The graphs of the functions Yv(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:
Modified Bessel's equation
The solution of this equation can be expressed through the so-called modified Bessel functions of the first and second kind:
Solution of the modified Bessel's equation
where Iv(x) and Kv(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:
Airy differential equation
It can be also reduced to the Bessel equation. Its solution is given by the Bessel functions of the fractional order :
solution of the Airy differential equation

3. The differential equation of type
differs from the Bessel equation only by a factor a2 before x2 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 n2 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.

   Example 1
Solve the differential equation  x2y'' + xy' + (3x2 − 2)y = 0.

Solution.
This equation has order √2 and differs from the standard Bessel equation only by factor 3 before x2. Therefore, the general solution of the equation is expressed by the formula
     
where C1, C2 are constants, J2(√3x) and Y2(√3x) are Bessel functions of the 1st and 2nd kind, respectively.

   Example 2
Solve the equation .

Solution.
This equation differs from the modified Bessel equation by factor 4 in front of x2. The order of the equation is v = 1/√2. Then the general solution is written through the modified Bessel functions in the following way:
     
where C1 and C2 are arbitrary constants.

   Example 3
Find the general solution of the differential equation  x2y'' + 2xy' + (x2 − 1)y = 0.

Solution.
We make the substitution:
     
Put these expressions back into the equation:
     
Indeed, we see that
     
Thus, the general solution for the function z(x) can be written in the form
     
Then the solution for the original function y(x) is given by
     
where C1 and C2 are arbitrary constants.

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