Bessel type differential equation

In summary, the conversation is about solving a Differential Equation with the form of a Bessel equation, but with an additional term that does not fit the Bessel equation. The person suggests a change of variable to try and transform the equation into a Bessel equation, but another person doubts that it is possible and suggests it could be transformed into a Kummer equation instead. A reference to a similar equation is also provided for further reading.
  • #1
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Homework Statement


Hello I am trying to solve the following Differential Equation:

[tex]r^2\frac{d^2R}{dr^2}+r\frac{dR}{dr}-\left[A^2r^4-B^2r^2-C^2]R=0[/tex]

where A,B and C are constants-

Homework Equations


I have read this equation is calle "Bessel wave eq" but I can't find the reference which is Moon and Spencer "Handbook"

The Attempt at a Solution


So with the change of variable z=r^2 and I get the following

[tex]4z^2\frac{d^2R}{dz^2}+4z\frac{dR}{dz}-\left[A^2z^2-B^2z-C^2]R=0[/tex]

which still doesn't have the form of a Bessel equation, because of the B^2z term!
How I get rid of this term? What change of variable can I make to get a Bessel equation?
Is there another way to solve this ODE?
 
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  • #2
What change of variable can I make to get a Bessel equation?
I should be surprised if this equation could be transformed into a Bessel equation.
Rather it could be changed into a Kummer equation (confluent hypergeometric)
 

1. What is a Bessel type differential equation?

A Bessel type differential equation is a type of second-order ordinary differential equation that involves a Bessel function, which is a special type of mathematical function that arises in many physical and engineering problems. It is named after the German mathematician Friedrich Bessel.

2. What are some applications of Bessel type differential equations?

Bessel type differential equations have numerous applications in physics, engineering, and other fields. They are commonly used to describe the behavior of waves, heat transfer, and electrical circuits. They also arise in problems involving cylindrical or spherical symmetry, such as in the study of fluid flow or celestial mechanics.

3. How do you solve a Bessel type differential equation?

The general solution to a Bessel type differential equation can be expressed in terms of Bessel functions, which can be found in tables or calculated using special software. In some cases, the equation can be simplified or transformed into a more familiar form, making it easier to solve. Specific boundary conditions are usually required to find the particular solution for a given problem.

4. Can Bessel type differential equations be solved numerically?

Yes, Bessel type differential equations can be solved numerically using numerical methods such as the Runge-Kutta method or finite difference methods. These methods involve approximating the solution at discrete points and using iterative algorithms to find a numerical solution.

5. Are there any special properties of Bessel functions that are useful in solving Bessel type differential equations?

Yes, Bessel functions have several important properties that make them useful in solving Bessel type differential equations. For example, they are orthogonal, meaning that they are perpendicular to each other when plotted on a graph. This property can be used to simplify the solution of certain problems. Bessel functions also have a recurrence relation, which allows for the calculation of higher-order Bessel functions from lower-order ones.

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