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MathGroup Archive 2000

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Re: Mod Bessel function bug ?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg23005] Re: [mg22988] Mod Bessel function bug ?
  • From: BobHanlon at aol.com
  • Date: Mon, 10 Apr 2000 02:22:32 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

DSolve[y''[x] - b^4*x^2*y[x] == 0, y[x], x]

{{y[x] -> Sqrt[x]*BesselI[-(1/4), (b^2*x^2)/2]*C[1] + 
     Sqrt[x]*BesselI[1/4, (b^2*x^2)/2]*C[2]}}

These two Bessel functions are different as shown by their plots

b = 10; xmax = 2/b;

Plot[{x^(1/2)*BesselI[-1/4, (b^2*x^2)/2], 
       x^(1/2)*BesselI[1/4, (b^2*x^2)/2]} , {x, 0, xmax}, 
    PlotStyle -> {RGBColor[1, 0, 0], RGBColor[0, 0, 1]}, PlotRange -> All];

Bob Hanlon

In a message dated 4/9/2000 2:02:32 AM, jrchaff at nwlink.com writes:

>I am trying to solve the differential equation,
>
>y''[x] - b^4*x^2*y[x]=0;
>
>Mathematica gives two indep solutions:
>
>y = c1*x^1/2*BesselI{-1/4,b^2*x^2/2}
>       + c2*x^1/2*BesselI{1/4,b^2*x^2/2};
>
>Now, since the order is not an integer, one would
>think that the first term, with order -1/4, is the same
>as the Modified Bessel "K" function, (times root x)
>so would go to zero with large x.
>
>However, plotting each term individually shows that
>Mathematica considers these two terms identical (!).
>So how can it claim they are independent solutions?
>
>Or am I making some mistake?
>


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