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Homework Help: Bessel's Eq. of order 0 and solution help

  1. Dec 12, 2005 #1
    Hello all,
    I'm studying for my diff/eq final, and I am having a lot of trouble understanding the answer to a question involving Bessel's equation of order 3/2. So, here's the question. please help::::::


    The following question concerns Bessel's Equation:

    [tex]x^2y''+xy'+(x^2-(\frac{3}{2})^2)y=0[/tex]

    The Bessel Function [tex]J_{3/2}(x)[/tex] is the Frobenius series solution which is finite at x = 0 Find the first three terms in its series expansion around x = 0.
    Ok, so I have the definition of the Bessel Function as:

    [tex]J_r(x)=\sum_{n=0}^{\infty} a_nx^{n+r}, a_0=1[/tex]

    where 'r' is the root of the indicial equation:

    [tex]r(r-1)+(1)r-(\frac{3}{2})^2[/tex]

    The answer to this question says::

    [tex]J_{3/2}=x^{3/2}(a_0+a_1x+a_2x^2+a_3x^3+...)[/tex]

    [tex]y=x^{3/2}(1+0x-\frac{1}{10}x^2+0x^3+...)[/tex]

    I don't know how they changed the [tex]a_0,a_1,a_2,...[/tex] values to those numbers right there.
    Thanks for the help.
    -Mark
     
    Last edited: Dec 12, 2005
  2. jcsd
  3. Dec 13, 2005 #2
    just insert the series [itex]J_{\frac{3}{2}}[/itex] on your O.D.E. The only way the equality will hold is if all the therms on the O.D.E. are zero. That will give you a condition over the [itex]a_n[/itex]'s.
     
    Last edited: Dec 13, 2005
  4. Dec 13, 2005 #3
    thx, Just one more question...

    thanks, incognitO. That actually helped me put things together. I just had one more question:::

    why does the term [tex]a_0[/tex] = 1??

    Is it just part of the definition of the Bessel Function, or something else?
     
  5. Dec 13, 2005 #4

    Dr Transport

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    Definition, if memory serves me correctly, the terms alternate. You can find the solution in terms of [tex] a_{n+2} = f(n)a_{n} [/tex].
     
  6. Dec 13, 2005 #5
    in series solutions with recurrence relations as the type mentioned by Dr Transport, is costumary to choose [itex]a_0=1[/itex], [itex]a_1=0[/itex] to find the first independent solution and then [itex]a_0=0[/itex] and [itex]a_1=1[/itex] to find the second independent solution. In the case of your Bessel function, this is not necesary scince [itex]J_{-3/2}[/itex] is also a solution. So the choice of [itex]a_0=1[/itex] comes from definition, wich makes Bessel functions special functions.
     
    Last edited: Dec 13, 2005
  7. Dec 15, 2005 #6
    thanks.

    Thanks incognitO, and Dr Transport. Really helped me with the understanding.

    -Mark
     
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