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# Vessel equation Questions watch

1. Hi my images sarey question and my , although I think I have gone wrong, can anyone help me out please?
2. Without beating through your working (which seems over-complex) we need to say that there is a solution to the Bessel equation such that C zero is non zero.

Once we have that solution written as a polynomial each coefficient in the polynomial must be separately zero.
i.e. if the solution is ax + bx^2 + cx^3 ... then a=0, b = 0, c=0 etc and each of a, b, c will be a polynomial in r.

So work out the bits (I write C rather than C sub zero to save having to latex this)

y=Cx^r, y'=Crx^(r-1) and y''=Cr(r-1)x^(r-2)

when these are substituted into Bessel's equn we get for terms involving C zero (again just using C). We are only interested in the C zero term

y(x) = Cx^r [ r (r-1) + r + (x^2 - m^2)] + other terms not involving C zero

Now multiplying this out (and rejecting the term in x^(r+2) cos that is not an x^r term) we get the coefficient of x^r is C[r^2 - r + r -m^2]

If C is non-zero then for the Solution to be valid the r term in the coefficient of x^r must be zero.

i.e. [r^2 - r + r -m^2] must be zero which gives us r^2=m^2

Hope that's clear
3. (Original post by nerak99)
Without beating through your working (which seems over-complex) we need to say that there is a solution to the Bessel equation such that C zero is non zero.

Once we have that solution written as a polynomial each coefficient in the polynomial must be separately zero.
i.e. if the solution is ax + bx^2 + cx^3 ... then a=0, b = 0, c=0 etc and each of a, b, c will be a polynomial in r.

So work out the bits (I write C rather than C sub zero to save having to latex this)

y=Cx^r, y'=Crx^(r-1) and y''=Cr(r-1)x^(r-2)

when these are substituted into Bessel's equn we get for terms involving C zero (again just using C). We are only interested in the C zero term

y(x) = Cx^r [ r (r-1) + r + (x^2 - m^2)] + other terms not involving C zero

Now multiplying this out (and rejecting the term in x^(r+2) cos that is not an x^r term) we get the coefficient of x^r is C[r^2 - r + r -m^2]

If C is non-zero then for the Solution to be valid the r term in the coefficient of x^r must be zero.

i.e. [r^2 - r + r -m^2] must be zero which gives us r^2=m^2

Hope that's clear
Thank you for this, I will give this a go
4. Suppose further that I wanted to do this;

All I can think of doing is just replacing "y" instead of "m" into the Bessel equation. Although that doesn't really help much.
5. bump
6. Well this falls out after some algebra.
Realise that the coefficients if x^(r+n) mainly involve Cn but the x^2-m^2 gives an -m^2 Cn x^(r+n) but the x^2 picks out C(n-2) x^(r+n).

When we multiply out the x^2 we end up with a term in x^(r+n+2) and so we have to pick out a term in Cn-2 to find the x^(r+n) term.

Sorry if that is not clear. Having a terrible time with getting tex to cooperate with subscripts
7. (Original post by nerak99)
Well this falls out after some algebra.
Realise that the coefficients if x^(r+n) mainly involve Cn but the x^2-m^2 gives an -m^2 Cn x^(r+n) but the x^2 picks out C(n-2) x^(r+n).

When we multiply out the x^2 we end up with a term in x^(r+n+2) and so we have to pick out a term in Cn-2 to find the x^(r+n) term.

Sorry if that is not clear. Having a terrible time with getting tex to cooperate with subscripts
Thank you I needed this tip will try and solve further later today.
Thank you I needed this tip will try and solve further later today.
You might also find this helpful:

http://mathworld.wolfram.com/FrobeniusMethod.html

The example they use on the method is the Bessel equation - lucky for you - but don't just copy it out or you won't learn anything...
9. (Original post by DFranklin)
You might also find this helpful:

http://mathworld.wolfram.com/FrobeniusMethod.html

The example they use on the method is the Bessel equation - lucky for you - but don't just copy it out or you won't learn anything...
great thanks, yes I will definietely use this and practice with TeeEm resources.

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