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

[zoki85]

$$y''+ \frac{1}{x}\cdot y' + (1-\frac{1}{x^2})\cdot y=0$$

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

 

[Chris Hillman]

Bessel equation, anyone?

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

 

[Matthew Rodman]

Yup. Have a look here.

 

[Chris Hillman]

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!

 

[zoki85]

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.

 

[momentum_waves]

$$y''+ \frac{1}{x}\cdot y' + (1-\frac{1}{x^2})\cdot y=0$$

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.

 

[Hurkyl]

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?

 

[momentum_waves]

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. :-)

 

[Chris Hillman]

Special functions are the applied mathematician's best friend!

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).

 

[momentum_waves]

Thanks, Chris & Hurkyl.

Can you perhaps recommend a suitable text on "special functions"? Thanks very much for your comments.

 

[Chris Hillman]

Some good books

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.

 

[DeadWolfe]

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.

 

[momentum_waves]

Thanks Chris & DeadWolfe for those excellent links.