Unusual Bessel-like Differential Equation with Negative Sign

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In summary, the conversation discusses a Bessel differential equation with a negative sign on the dependent variable and the attempt to transform it into a modified Bessel differential equation by multiplying it with x^2. The person is seeking assistance in understanding the equation and solving it on their own. A resource for the modified spherical Bessel differential equation is provided.
  • #1
maistral
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Hi. I have this Bessel look-a-like DE (however it has the negative sign on the dependent variable):

2pql7i8.png


My attempt at a solution is by (obviously) trying to transform it into a Bessel DE by multiplying it with x2 on both sides, then it would look like the Spherical Bessel differential equation, except that there's that negative sign beside y.

Then what? :nb) I have no idea what to do. Can someone tell me what this DE is supposed to look like so I can solve it on my own?
 
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  • #2
maistral said:
Hi. I have this Bessel look-a-like DE (however it has the negative sign on the dependent variable):

View attachment 203702

My attempt at a solution is by (obviously) trying to transform it into a Bessel DE by multiplying it with x2 on both sides, then it would look like the Spherical Bessel differential equation, except that there's that negative sign beside y.

Then what? :nb) I have no idea what to do. Can someone tell me what this DE is supposed to look like so I can solve it on my own?
http://mathworld.wolfram.com/ModifiedSphericalBesselDifferentialEquation.html
 
  • #3
Got it. Thanks!
 

1. What is a "Weird looking Bessel-like DE"?

A "Weird looking Bessel-like DE" refers to a differential equation that resembles a Bessel function, but with unusual or unexpected features. Bessel functions are a type of special function that often arise in physics and engineering problems involving circular or cylindrical symmetry.

2. How is a "Weird looking Bessel-like DE" different from a regular Bessel function?

A "Weird looking Bessel-like DE" may have different coefficients or boundary conditions compared to a regular Bessel function. Additionally, it may have additional terms or unusual features that make it unique from traditional Bessel functions.

3. What applications use "Weird looking Bessel-like DE"?

"Weird looking Bessel-like DE" equations have been used in a variety of scientific fields, including acoustics, electromagnetics, and fluid dynamics. They are particularly useful in problems involving circular or cylindrical symmetry, such as in the design of resonators or waveguides.

4. How do you solve a "Weird looking Bessel-like DE"?

Solving a "Weird looking Bessel-like DE" typically involves using specialized techniques, such as Frobenius series or integral transforms, to find a closed form solution. In some cases, numerical methods may be required. It is important to carefully consider the boundary conditions and physical interpretation of the solution when solving these types of equations.

5. What are some real-world examples of "Weird looking Bessel-like DE"?

Some examples of real-world applications of "Weird looking Bessel-like DE" equations include the analysis of sound propagation in curved pipes, the design of microwave waveguides, and the study of vortex shedding in fluid flow. They have also been used in the analysis of resonant structures in musical instruments and in the design of optical fibers.

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