Re: DSolve:: Bessel's differential equation
- To: mathgroup at smc.vnet.net
- Subject: [mg64277] Re: DSolve:: Bessel's differential equation
- From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
- Date: Thu, 9 Feb 2006 02:44:43 -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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