Can Bessel's equation be solved using only Frobenius method?

[A Malik]

can we use only frobenius method to solve bessel equation?

 

[JJacquelin]

can we use only frobenius method to solve bessel equation?

Why do you want to use series developments to solve Bessel ODEs ?
On the contrary, the Bessel functions are closed forms which avoid the series developments.

 

[jackmell]

Why do you want to use series developments to solve Bessel ODEs ?
On the contrary, the Bessel functions are closed forms which avoid the series developments.

That's confussing. I mean I look in my DE book, even gotta' section on Bessel DE, and bam! Series solutions.

 

[JJacquelin]

That's confussing. I mean I look in my DE book, even gotta' section on Bessel DE, and bam! Series solutions.

Sorry, I cannot understand what is the confusion.
For example, in attachment, two ODEs are compared.
Both solutions can be expressed on closed form thanks to functions which are also infinite series. Of course, the name given to the functions are not the same : J0(x) and cos(x) for example. But I cannot see any other difference, except that cos(x) is more popular than J0(x). But popularity has nothing to do with maths.

 

[jackmell]

Sorry, I cannot understand what is the confusion.
For example, in attachment, two ODEs are compared.
Both solutions can be expressed on closed form thanks to functions which are also infinite series. Of course, the name given to the functions are not the same : J0(x) and cos(x) for example. But I cannot see any other difference, except that cos(x) is more popular than J0(x). But popularity has nothing to do with maths.

Ain't there suppose to be a factor of 2^{2k} in there?

May I also ask how is that first equation solved directly if one does not know before-hand, it is a Bessel DE? I think we'd have to resort to power series. However, the OP I think is asking, is there another way other than power series to solve it?

By the way, thanks for helping us in here as you solve in an elegant manner, equations I can't solve and maybe others too. :)

 

[A Malik]

yes there must be a factor
[2]^2k in denominator which will distinguish it from cos(x) series
yup... i was asking any other method to get solution of Bessel equation...
If we use power series method to solve it
what we have to suppose y(x)= ?

 

[JJacquelin]

Ain't there suppose to be a factor of 2^{2k} in there?

May I also ask how is that first equation solved directly if one does not know before-hand, it is a Bessel DE? I think we'd have to resort to power series. However, the OP I think is asking, is there another way other than power series to solve it?

By the way, thanks for helping us in here as you solve in an elegant manner, equations I can't solve and maybe others too. :)

You are right, I forgot the factor 2^{2k}. It is corrected below. Thank you for the remark.

if one does not know how to recognize a Bessel ODE, he cannot solve it directly. More over, if he doesn't know what are the first and second derivatives of the Bessel functions, he cannot put them back in the ODE and see if the equation is satisfied. Moreover and moreover, if he doesn'nt know the DE of the Bessel function, he has no chance to recognize it into the power series of the solutions of the ODE. That is why the best way is to acquire the ability to recognize if an ODE is a Bessel ODE or not.

But it is the same for the ODEs of the kind y''+y=0 :
if one does not know how to recognize a circular ODE, he cannot solve it directly. More over, if he doesn't know what are the first and second derivatives of trigonometric functions, he cannot put them back in the ODE and see if the equation is satisfied. Moreover and moreover, if he doesn'nt know the DE of the trigonometric functions, he has no chance to recognize it into the power series of the solutions of the ODE. That is why the best way is to acquire the ability to recognize if an ODE is a circular ODE or not.
The difference is that trigonometric functions are well known, that ODEs of the kind y''+y=0 are well known and that related background is known before hand, which is generally not the case for ODEs of the Bessel kind. The difference is that many people already aquiered the ability to recognize if an ODE is a circular ODE, but few have learned how to recognize a Bessel ODE.

 

[HallsofIvy]

If you are referring to solving differential equations in terms of "elementary functions" that cannot be done so then, yes, you would have to use power series. However, it is very simple to solve Bessel's equations in terms of Bessel Functions that are defined as independent solutions to Bessel's equations.