Thread: Bessel differencial equation

please please can someone hep me on this as i dont have a clue where to start. I know its a ,ong question but if someone coul do itthen would really give me a bse to understand the topic

thanks loads
Edgar

Show that Jn(px) satisfies

x^2y'' + xy' + (p^2x^2 - n^2)y = 0;

and deduce

[x d/dx Jn(px)] (Subscript)x + ((p^2)x -n^2/x) Jn(px) = 0:

Show that the integral between l and 0 of xJn(px)Jn(qx) dx =

l/(q^2 - p^2) [pJn(ql)J'n(pl) - qJn(pl)J'n(ql)] ;

and, using l'Hopital's rule,

the integral between l and 0 of
xJn^2(px) dx =l^2/2 Jn'^2(pl) + 1 - (n^2/p^2l^2)Jn^2(pl)

Deduce that, if l is such that Jn(pl), Jn(ql) are both zero (i.e. pl and ql are both
roots of Jn(x)), then
the integarl between l and 0 of
xJn(px)Jn(qx) dx = 0; p doesnt equal q

Originally Posted by edgar davids

please please can someone hep me on this as i dont have a clue where to start. I know its a ,ong question but if someone coul do itthen would really give me a bse to understand the topic

thanks loads
Edgar

Show that Jn(px) satisfies

x^2y'' + xy' + (p^2x^2 - n^2)y = 0;

What is "Jn(px)"?

The equation,

Look like an analogue of the Bessel equation.
I can show that there exists a solution satisfiying the Frobenius method by shows that these functions are Legendre polynomials of orders 1 and 2. If that is what you are asking.

Jn(x) is a series solution to bessels equation

Originally Posted by edgar davids

Jn(x) is a series solution to bessels equation

Okay.

Given, the Bessel equation,

The Bessel function of order ,
satisfies the differencial equation.

We need to show that,
satisfies the differencial equation.
(*)

Since,

We have that,



With the boxed equation we can show that satisfies the differencial equation (*).

Note that,




Substitute that into (*) to get,

And this expression is equal to zero by the boxed equation.
Hence satisfies (*).