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MathGroup Archive 1998

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Re: ODE

  • To: mathgroup at smc.vnet.net
  • Subject: [mg14532] Re: ODE
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Thu, 29 Oct 1998 04:33:28 -0500
  • Organization: University of Western Australia
  • References: <70rapb$sg7@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Ing. Alessandro Toscano Dr. wrote:

> I found that the ODE:
> 
> (A+B*x)*y(x) - y'(x)/y + y''(x)=0
> 
> has an analytical, series solution near a regular singular point.
> 
> Is there any package or notebook which solves ODE in terms of the so
> called series solutions?

As a concrete example, consider the Bessel differential equation:

In[1]:= beseqn = x^2 y''[x] + x y'[x] + (x^2 - n^2) y[x]

The point x=0 is a regular singular point.  You can use the indicial
equation to determine the exponents of the singularity:
 
In[2]:= beseqn /. y -> Function[x, a[k]*x^k] In[3]:= Collect[%, x,
Factor]
In[4]:= Simplify[% /. 
   (c_.) a[k] x^(k + (m_.)) -> 
    (c a[k] x^(k + m) /. k -> k - m)]

We now have the indicial equation which has roots k = +/-n. Now we can
determine the series solution.  First we simplify the differential
equation:

In[5]:= Simplify[beseqn/x^n /. y -> Function[{x}, x^n g[x]]]

Then we expand into a series (g[0] -> 1 without loss of generality)

In[6]:= % + O[x]^7 /. g[0] -> 1

and solve for the undetermined coefficients:

In[7]:= Solve[% == 0, Union[Cases[%, Derivative[_][g][_], Infinity]]]

Here is the resulting series expansion:

In[8]:= x^n (g[x] + O[x]^7 /. g[0] -> 1 /. First[%])

which can be compared with the built-in series:

In[9]:= FunctionExpand[Gamma[n + 1] 2^n * Series[BesselJ[n, x], {x, 0,
7}]]

Cheers,
	Paul

____________________________________________________________________ 
Paul Abbott                                   Phone: +61-8-9380-2734
Department of Physics                           Fax: +61-8-9380-1014
The University of Western Australia            Nedlands WA  6907       
mailto:paul at physics.uwa.edu.au  AUSTRALIA                       
http://www.physics.uwa.edu.au/~paul

            God IS a weakly left-handed dice player
____________________________________________________________________


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