Is the DE y''+ (1/x)y' + (1-1/x^2)y=0 solvable with explicit methods?

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Discussion Overview

The discussion centers around the differential equation y'' + (1/x)y' + (1 - 1/x^2)y = 0, exploring its solvability using explicit methods. Participants consider various approaches, including connections to Bessel functions and numerical methods.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant suggests that the equation appears simple but may be unsolvable explicitly, seeking recommendations.
  • Another participant identifies the equation as a form of the Bessel equation for n=1, proposing solutions involving Bessel functions J_1 and Y_1.
  • A later reply mentions solving the equation in the form of an infinite series, noting its correspondence to Bessel functions but expressing a desire for a more explicit solution.
  • Some participants propose numerical methods, such as reducing the ODE to two first-order ODEs and applying a shooting algorithm, while acknowledging that Bessel function solutions can become complex and often lead to numerical solutions.
  • There is a discussion about the exotic nature of Bessel functions compared to trigonometric functions, with some participants noting the need to consult tables for specific solutions.
  • Participants highlight the utility of special functions in applied mathematics and suggest that modern computer algebra systems can assist in working with these functions.
  • Several participants recommend various texts on special functions, emphasizing that good discussions can be found in mathematical methods books suitable for different educational levels.

Areas of Agreement / Disagreement

Participants express differing views on the explicit solvability of the differential equation, with some suggesting it may not be solvable explicitly while others point to established solutions involving Bessel functions. The discussion remains unresolved regarding the best approach to take.

Contextual Notes

Some limitations include the potential complexity of Bessel function solutions, the dependence on specific boundary conditions for numerical methods, and the varying levels of familiarity with special functions among participants.

Who May Find This Useful

This discussion may be of interest to students and professionals in mathematics, physics, and engineering, particularly those exploring differential equations and special functions.

zoki85
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[tex]y''+ \frac{1}{x}\cdot y' + (1-\frac{1}{x^2})\cdot y=0[/tex]

Looks simple but it's a trouble.Probably unsolvable (explicitely).
Recommendation?
 
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Bessel equation, anyone?

Isn't that the Bessel equation for the case [itex]n=1[/itex]? Where y is dependent variable and x is independent variable? (Multiply by [itex]x^2[/itex].) Solutions
[tex] y(x) = c_1 \, J_1(x) + c_2 \, Y_1(x)[/tex]
where [itex]J_1, \, Y_1[/itex] are Bessel functions. This equation and its solutions are discussed in all good Mathematical Methods textbooks.
 
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Iconic pedagogical websites

eqworld is a great website which everyone should bookmark. I have done that myself so I don't know why I didn't think of mentioning it!
 
Thanks both of you .I didn't know of the name .
Well,I should have said that I solved it in form of infinite series .Now I see it correspond to Bessel function.I hoped that there could be something nicer and more explicite.
 
zoki85 said:
[tex]y''+ \frac{1}{x}\cdot y' + (1-\frac{1}{x^2})\cdot y=0[/tex]

Looks simple but it's a trouble.Probably unsolvable (explicitely).
Recommendation?

You can also try solving it numerically by reducing the ODE into two first order ODE's, then apply a suitable 'shooting type' algorithm if you have suitable boundary conditions. This can be very informative.

These Bessel function solutions generally end up becoming pretty messy & often result in nothing other than a glorified numerical solution, anyway. Matlab is a great simulation platform.
 
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momentum_waves said:
These Bessel function solutions generally end up becoming pretty messy & often result in nothing other than a glorified numerical solution, anyway.
The Bessel functions are well studied, aren't they? How is it much different than, for example, when the solution is a trigonometric function?
 
Bessel functions are a little more exotic aren't they? Often we have to consult tables, unless the solutions are readily available.

Determining the specific solutions from the general solutions can be little more tricky. We use these methods fairly often in heat-transfer work, for instance. It often turns out to be far simpler to use numerical techniques. :-)
 
Special functions are the applied mathematician's best friend!

Hurkyl said:
The Bessel functions are well studied, aren't they? How is it much different than, for example, when the solution is a trigonometric function?

Ditto Hurkyl. momentum_waves, if you've never studied a book on "special functions", this is a wonderful topic. Modern computer algebra systems incorporate a large store of knowledge about special functions and can efficiently convert between them, although with nowhere near the proficiency of the best human experts (so far).
 
  • #10
Thanks, Chris & Hurkyl.

Can you perhaps recommend a suitable text on "special functions"? Thanks very much for your comments.
 
  • #11
Some good books

momentum_waves said:
Can you perhaps recommend a suitable text on "special functions"?

Well, I am not sure I would recommend a specialist text for all readers. A good discussion of the most important special functions is contained in good books on mathematical methods. Second or third year undergraduates can see:

Mary L. Boas, Mathematical methods in the physical sciences. 3rd Edition. Wiley, 2006.

This book offers, I think, a very tasteful selection of material given the limitations of space.

Seniors can see:

Harold Jeffreys and Bertha Swiles Jeffreys, Methods of Mathematical Physics. 3rd Edition. Cambridge University Press, 1953 (reprinted 1972).

(Unfortunately, J&J use a somewhat idiosyncratic notation which might hamper comparision with other books. OTH, one could argue that this is a perennial problem and students may as well encounter it sooner rather than later.)

A more computational book I really like, which offers lots of valuable snippets (but is perhaps not so easy to use as a reference), is:

Derek Richards, Advanced Mathematical Methods with Maple, Cambridge University Press, 2002.

A good graduate level textbook is

George E. Andrews, Richard Askey, and Ranjan Roy, Special Functions, Cambridge University Press, 2000.

No doubt every physicist, mathematician, or engineer has on their shelves a copy of

Abramowitz and Stegun, Handbook of Mathematical Functions, National Bureau of Standards, 1972

Probably cited more often than the most religious texts.
 
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  • #12
Watson wrote a great book called "A Treatise on the Theory of Bessel Functions" which has more facts about them than anybody would ever care to know.
 
  • #13
Thanks Chris & DeadWolfe for those excellent links.
 

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