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Bessel Differential Equation
The linear second order ordinary differential equation of type ${x^2}y'' + xy' = \left( {{x^2} - {v^2}} \right)y = 0$ 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 $y\left( x \right) = {C_1}{J_v}\left( x \right) + {C_2}{J_{ - v}}\left( x \right),$ where $${C_1},$$ $${C_2}$$ are arbitrary constants and $${J_v}\left( x \right),$$ $${J_{ - v}}\left( x \right)$$ 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: ${J_v}\left( x \right) = \sum\limits_{p = 0}^\infty {\frac{{{{\left( { - 1} \right)}^p}}}{{\Gamma \left( {p + 1} \right)\Gamma \left( {p + v + 1} \right)}}{{\left( {\frac{x}{2}} \right)}^{2p + v}}} .$ The Gamma function is the generalization of the factorial function from integers to all real numbers. It has, in particular, the following properties: ${\Gamma \left( {p + 1} \right) = p!,}\;\; {\Gamma \left( {p + v + 1} \right) = \left( {v + 1} \right)\left( {v + 2} \right) \cdots \left( {v + p} \right)\Gamma \left( {v + 1} \right).}$ The Bessel functions of the negative order ($$-v$$) (assuming that $$v > 0$$) are written in similar way: ${J_{ - v}}\left( x \right) = \sum\limits_{p = 0}^\infty {\frac{{{{\left( { - 1} \right)}^p}}}{{\Gamma \left( {p + 1} \right)\Gamma \left( {p - v + 1} \right)}}{{\left( {\frac{x}{2}} \right)}^{2p - v}}} .$ The Bessel functions can be calculated in most mathematical software packages. For example, the Bessel functions of the $$1$$st kind of orders $$v = 0$$ to $$v = 4$$ are shown in Figure $$1.$$ The corresponding functions are also available in MS Excel.
 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 $${J_v}\left( x \right)$$ and $${J_{ - v}}\left( x \right)$$ can become dependent from each other. In this case the general solution is described by another formula: $y\left( x \right) = {C_1}{J_v}\left( x \right) + {C_2}{Y_v}\left( x \right),$ where $${Y_v}\left( x \right)$$ 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 $${Y_v}\left( x \right)$$ can be expressed through the Bessel functions of the first kind $${J_v}\left( x \right)$$ and $${J_{ - v}}\left( x \right):$$ ${Y_v}\left( x \right) = \frac{{{J_v}\left( x \right)\cos \pi v - {J_{ - v}}\left( x \right)}}{{\sin \pi v}}.$ The graphs of the functions $${Y_v}\left( x \right)$$ 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 $$x$$ with $$-ix.$$ This equation has the form: ${x^2}y'' + xy' - \left( {{x^2} + {v^2}} \right)y = 0.$ The solution of this equation can be expressed through the so-called modified Bessel functions of the first and second kind: ${y\left( x \right) = {C_1}{J_v}\left( { - ix} \right) + {C_2}{Y_v}\left( { - ix} \right) } = {{C_1}{I_v}\left( x \right) + {C_2}{K_v}\left( x \right),}$ where $${I_v}\left( x \right)$$ and $${K_v}\left( x \right)$$ are modified Bessel functions of the $$1$$st and $$2$$nd kind, respectively.

2. The Airy differential equation known in astronomy and physics has the form: $y'' - xy = 0.$ It can be also reduced to the Bessel equation. Its solution is given by the Bessel functions of the fractional order $$\pm {\large\frac{1}{3}\normalsize}:$$ ${y\left( x \right) } = {{C_1}\sqrt x {J_{\large\frac{1}{3}\normalsize}}\left( {\frac{2}{3}i{x^{\large\frac{3}{2}\normalsize}}} \right) + {C_2}\sqrt x {J_{ - \large\frac{1}{3}\normalsize}}\left( {\frac{2}{3}i{x^{\large\frac{3}{2}\normalsize}}} \right).}$
3. The differential equation of type ${x^2}y'' + xy' + \left( {{a^2}{x^2} - {v^2}} \right)y = 0$ differs from the Bessel equation only by a factor $${a^2}$$ before $${x^2}$$ and has the general solution in the form: $y\left( x \right) = {C_1}{J_v}\left( {ax} \right) + {C_2}{Y_v}\left( {ax} \right).$
4. The similar differential equation ${x^2}y'' + axy' + \left( {{x^2} - {v^2}} \right)y = 0$ is reduced to the Bessel equation ${x^2}z'' + xz' + \left( {{x^2} - {n^2}} \right)z = 0$ by using the substitution $y\left( x \right) = {x^{\large\frac{{1 - a}}{2}\normalsize}}z\left( x \right).$ Here the parameter $${n^2}$$ denotes: ${n^2} = {v^2} + \frac{1}{4}{\left( {a - 1} \right)^2}.$ As a result, the general solution of the differential equation is given by $y\left( x \right) = {x^{\large\frac{{1 - a}}{2}\normalsize}}\left[ {{C_1}{J_n}\left( x \right) + {C_2}{Y_n}\left( x \right)} \right].$
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 $${x^2}y'' + xy' + \left( {3{x^2} - 2} \right)y = 0.$$

Solution.
This equation has order $$\sqrt 2$$ and differs from the standard Bessel equation only by factor $$3$$ before $${x^2}.$$ Therefore, the general solution of the equation is expressed by the formula $y\left( x \right) = {C_1}{J_{\sqrt 2 }}\left( {\sqrt 3 x} \right) + {C_2}{Y_{\sqrt 2 }}\left( {\sqrt 3 x} \right),$ where $${C_1},$$ $${C_2}$$ are constants, $${J_{\sqrt 2 }}\left( {\sqrt 3 x} \right)$$ and $${Y_{\sqrt 2 }}\left( {\sqrt 3 x} \right)$$ are Bessel functions of the $$1$$st and $$2$$nd kind, respectively.

Example 2
Solve the equation $${x^2}y'' + xy' - \left( {4{x^2} + {\large\frac{1}{2}\normalsize}} \right)y = 0.$$

Solution.
This equation differs from the modified Bessel equation by factor $$4$$ in front of $${x^2}.$$ The order of the equation is $$v = {\large\frac{1}{{\sqrt 2 }}\normalsize}.$$ Then the general solution is written through the modified Bessel functions in the following way: $y\left( x \right) = {C_1}{I_{\large\frac{1}{{\sqrt 2 }}\normalsize}}\left( {2x} \right) + {C_2}{K_{\large\frac{1}{{\sqrt 2 }}\normalsize}}\left( {2x} \right),$ where $${C_1}$$ and $${C_2}$$ are arbitrary constants.

Example 3
Find the general solution of the differential equation $${x^2}y'' + 2xy' + \left( {{x^2} - 1} \right)y = 0.$$

Solution.
We make the substitution: ${y = {x^{\large\frac{{1 - 2}}{2}\normalsize}}z = {x^{ - \large\frac{1}{2}\normalsize}}z,}\;\; {\Rightarrow y' = - \frac{1}{2}{x^{ - \large\frac{3}{2}\normalsize}}z + {x^{ - \large\frac{1}{2}\normalsize}}z',}\;\; {\Rightarrow y'' = \frac{3}{4}{x^{ - \large\frac{5}{2}\normalsize}}z - \frac{1}{2}{x^{ - \large\frac{3}{2}\normalsize}}z' - \frac{1}{2}{x^{ - \large\frac{3}{2}\normalsize}}z' + {x^{ - \large\frac{1}{2}\normalsize}}z'' } = {\frac{3}{4}{x^{ - \large\frac{5}{2}\normalsize}}z - {x^{ - \large\frac{3}{2}\normalsize}}z' + {x^{ - \large\frac{1}{2}\normalsize}}z''.}$ Put these expressions back into the equation: ${x^2}y'' + 2xy' + \left( {{x^2} - 1} \right)y = 0,$ ${\Rightarrow {x^2}\left( {\frac{3}{4}{x^{ - \large\frac{5}{2}\normalsize}}z - {x^{ - \large\frac{3}{2}\normalsize}}z' + {x^{ - \large\frac{1}{2}\normalsize}}z''} \right) } + {2x\left( { - \frac{1}{2}{x^{ - \large\frac{3}{2}\normalsize}}z + {x^{ - \large\frac{1}{2}\normalsize}}z'} \right) } + {\left( {{x^2} - 1} \right){x^{ - \large\frac{1}{2}\normalsize}}z = 0,}$ ${\Rightarrow \color{blue}{\frac{3}{4}{x^{ - \large\frac{1}{2}\normalsize}}z} } - {\color{red}{{x^{ - \large\frac{1}{2}\normalsize}}z'} + {x^{\large\frac{3}{2}\normalsize}}z'' } - {\color{blue}{{x^{ - \large\frac{1}{2}\normalsize}}z} + \color{red}{2{x^{\large\frac{1}{2}\normalsize}}z'} } + {\color{blue}{{x^{\large\frac{3}{2}\normalsize}}z} - \color{blue}{{x^{ - \large\frac{1}{2}\normalsize}}z} = 0,}$ $\Rightarrow \left. {{x^{\large\frac{3}{2}\normalsize}}z'' + \color{red}{{x^{ - \large\frac{1}{2}\normalsize}}z'} + \color{blue}{\left( { - \frac{5}{4}{x^{ - \large\frac{1}{2}\normalsize}} + {x^{\large\frac{3}{2}\normalsize}}} \right)z} = 0} \right| \cdot {x^{\large\frac{1}{2}\normalsize}},$ $\Rightarrow {x^2}z'' + xz' + \left( {{x^2} - \frac{5}{4}} \right)z = 0.$ Indeed, we see that ${{n^2} = {v^2} + \frac{1}{4}{\left( {a - 1} \right)^2} } = {1 + \frac{1}{4}{\left( {2 - 1} \right)^2} } = {1 + \frac{1}{4} } = {\frac{5}{4}.}$ Thus, the general solution for the function $$z\left( x \right)$$ can be written in the form $z\left( x \right) = {C_1}{J_{\large\frac{{\sqrt 5 }}{2}\normalsize}}\left( x \right) + {C_2}{Y_{\large\frac{{\sqrt 5 }}{2}\normalsize}}\left( x \right).$ Then the solution for the original function $$y\left( x \right)$$ is given by ${y\left( x \right) = {x^{ - \large\frac{1}{2}\normalsize}}z\left( x \right) } = {\frac{1}{{\sqrt x }}\left[ {{C_1}{J_{\large\frac{{\sqrt 5 }}{2}\normalsize}}\left( x \right) + {C_2}{Y_{\large\frac{{\sqrt 5 }}{2}\normalsize}}\left( x \right)} \right],}$ where $${C_1}$$ and $${C_2}$$ are arbitrary constants.