Solving Bessel ODE by Breaking into First Order Equations

However, the difficulty arises when trying to implement the 1/x term in the second equation. This can be solved by using a transformation to convert the equation into a form that can be solved by the ODE solver.
  • #1
jschmid2
6
0
Homework Statement [/b]
I would like to solve a second order ODE, by breaking it up into two first order ODES. However, I'm having a bit of difficultly here.

The ODE is [tex]y''=-(\frac{1}{x}y'+y)[/tex]
and the true solution is Bessel of first kind, order zero.

I have tried setting z=y' which means
z'=y''=the equation shown above.

However, I am having difficultly implementing the [tex]\frac{1}{x}[/tex]

I'm trying to program it in matlab, but none of the ODE solvers knows how to deal with the 1/x.

Any ideas?
 
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  • #2
Homework Equationsy''=-(\frac{1}{x}y'+y)The Attempt at a SolutionThe way to solve this problem is to define a new variable z = y', and then solve the following two first order differential equations: y'=z z'=-(\frac{1}{x}z+y) Then, using an ODE solver such as Matlab's ode45, you can solve these two equations simultaneously.
 

Related to Solving Bessel ODE by Breaking into First Order Equations

1. What is a Bessel ODE math problem?

A Bessel ODE (ordinary differential equation) math problem is a type of mathematical equation that involves finding the solution for a function in terms of its derivatives. These equations are named after the mathematician Friedrich Bessel and are commonly used in the study of vibrations, heat transfer, and other physical phenomena.

2. What is the history behind the Bessel ODE?

The Bessel ODE was first introduced by Friedrich Bessel in the 19th century while studying the motion of a vibrating circular membrane. He discovered that the solution to this problem could be expressed in terms of a special type of mathematical function now known as the Bessel function.

3. What are some real-world applications of the Bessel ODE?

The Bessel ODE is used in many fields of science and engineering, including acoustics, electromagnetics, fluid dynamics, and quantum mechanics. It is particularly useful in solving problems involving circular symmetry, such as the vibration of circular membranes, heat transfer in cylindrical objects, and the behavior of electromagnetic waves in circular waveguides.

4. How is the Bessel ODE solved?

The Bessel ODE can be solved using various techniques, such as power series, Frobenius method, or Laplace transform. The choice of method depends on the specific form of the equation and the boundary conditions. In some cases, the solution may also involve using tables of Bessel functions or numerical methods.

5. What are some challenges when solving Bessel ODE math problems?

One of the main challenges in solving Bessel ODE math problems is identifying the appropriate method for solving the equation and applying it correctly. These equations can also have multiple solutions, making it important to verify the results. Additionally, the presence of complex numbers in the solutions can make it challenging to interpret the physical meaning of the results.

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