       DSolve:: Bessel's differential equation

• To: mathgroup at smc.vnet.net
• Subject: [mg64261] DSolve:: Bessel's differential equation
• From: bd satish <bdsatish at gmail.com>
• Date: Wed, 8 Feb 2006 03:53:58 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```    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)?

```

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