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MathGroup Archive 2006

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Re: DSolve:: Bessel's differential equation

  • To: mathgroup at smc.vnet.net
  • Subject: [mg64265] Re: DSolve:: Bessel's differential equation
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Wed, 8 Feb 2006 06:29:49 -0500 (EST)
  • Organization: The Open University, Milton Keynes, UK
  • References: <dsccr4$pb1$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

bd satish wrote:
>     Hi buddies,
> 
>                Here is a differential equation , which could not be done by
> DSolve  (in Version 5.0 ).
> This occurs in the mathematical modelling of a simple pendulum of length L
> and a parameter  k .
> (Actually , k ^ 2 = frequency of oscillation^2 / acceleration due to gravity
> )
> 
>             (L - x) y''[x] - y'[x] + k^2 y[x] == 0         .... (1)
> 
>   The above equation is in fact reducible to Bessel's differential equation
> (with order n = 0 )
> 
> with the substituions  L-x = z and s = 2 k Sqrt[z]
> 
>              y''[s] + 1 /s  y'[s] + y[s] ==0              .... (2)
> 
>  The text-book says that the solution of eqn (1)  contains a BesselJ[0,2 k
> Sqrt[L-x] ].
> 
> 
>   How can I get DSolve to answer (1) directly , without resorting to eqn (2)?
> 
> 
It works with Mathematica 5.2:

In[1]:=
eqn = (L - x)*Derivative[2][y][x] - Derivative[1][y][x] + k^2*y[x] == 0;

In[2]:=
DSolve[eqn, y, x]

Out[2]=
{{y -> Function[{x}, (Sqrt[-2*L + 2*x]*BesselI[0, 2*Sqrt[k^2*(-L + 
x)]]*C[1])/Sqrt[2*L - 2*x] +
       (Sqrt[-2*L + 2*x]*BesselK[0, 2*Sqrt[k^2*(-L + x)]]*C[2])/Sqrt[2*L 
- 2*x]]}}

In[3]:=
$Version

Out[3]=
"5.2 for Microsoft Windows (June 20, 2005)"

Best regards,
/J.M.


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