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Solutions to 2 similar ODE's. Why does they differ that much?

  1. Nov 18, 2012 #1

    fluidistic

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    I'm reading on wikipedia (http://en.wikipedia.org/wiki/Vibrations_of_a_circular_drum#The_radially_symmetric_case) that the solution to the ODE [itex]R''+(1/r)R'-KR=0[/itex], by letting [itex]K=-\lambda ^2[/itex] is a linear combination of the Bessel functions of order 0, namely [itex]R(r)=c_1 J_0 (\lambda r ) + c_2 Y_0 (\lambda r)[/itex]. Then they say that c_2 must equal 0 for the solution to make physical sense (because Y_0 diverges at r=0) and thus the solution to the ODE is of the form [itex]R(r)=c_1J_0 (\lambda r )[/itex]. So far so good.
    However in the problem of my thread https://www.physicsforums.com/showthread.php?t=652277, I had the equation [itex]R''+\frac{2R'}{r}+CR=0[/itex] which is very similar to the ODE in wikipedia and I've determined that the physically acceptable solution is of the form [itex]R(r)=c_2\frac{\sin (\sqrt C r )}{r}[/itex] (I've checked that it indeed satisfies the ODE). I find this unbelievable that the solution to the ODE of my problem differs so much from the one in wikipedia. What's going on here? I'm totally clueless.

    Edit: Here's a graph between J_0(x) and sin (x) /x : http://www.wolframalpha.com/input/?i=graph+J_0+(x)+and+sin+(x)+/x+from+0+to+100. They are very similar! Is it possible to write the Bessel function of the first kind of order 0 as a constant times sin (kx) /x?
     
    Last edited: Nov 18, 2012
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  3. Nov 18, 2012 #2

    haruspex

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    According to http://en.wikipedia.org/wiki/Bessel_function, "The graphs of Bessel functions look roughly like oscillating sine or cosine functions that decay proportionally to 1/√x, although their roots are not generally periodic, except asymptotically for large x."
     
  4. Nov 18, 2012 #3

    fluidistic

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    Ah thank you! I missed this part!
    So basically although the solutions to [itex]R''+(1/r)R'-KR=0[/itex] and [itex]R''+(2/r)R'-KR=0[/itex] can look similar, there's still some "huge" difference in that Bessel functions are involved in the first ODE but not in the second. And that we cannot write the Bessel function of the first kind of order 0 as a constant times sin (kx) /x.
    Wow, impressive.

    P.S.:I'm just seeing http://en.wikipedia.org/wiki/Spherical_Bessel_function#Spherical_Bessel_functions:_jn.2C_yn. Apparently [itex]\sin x /x[/itex] is a spherical Bessel function (of order 0 I think)!!! That appears when solving the Helmholtz equation in spherical coordinates. Wow.
     
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