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MathGroup Archive 1998

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Re: Way to evaluate D[(1-x^2)y''[x],{x,n}] ?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg14938] Re: [mg14914] Way to evaluate D[(1-x^2)y''[x],{x,n}] ?
  • From: Jurgen Tischer <jtischer at col2.telecom.com.co>
  • Date: Fri, 27 Nov 1998 03:49:35 -0500
  • Organization: Universidad del Valle
  • References: <199811252248.RAA26500@smc.vnet.net.>
  • Sender: owner-wri-mathgroup at wolfram.com

Hi Phillip,
I can't imagine how to make Mathematica calculating a general formula
for the nth derivative, but if you find it you can use Mathematica to
proof it by induction:

This is the formula found by inspection.

In[1]:= eq[n_] = -(n - 1)*n*D[y[x], {x, n}] - 
   2*n*x*D[Derivative[1][y][x], {x, n}] + 
   (1 - x^2)*D[Derivative[1][Derivative[1][y]][x], {x, n}];

Now 

In[2]:= eq[1] == D[(1 - x^2)*Derivative[1][Derivative[1][y]][x], x]

Out[2]= True

In[3]:= eq[n+1]==D[eq[n],x]//Simplify

Out[3]= True

Jurgen

Dr Phillip Kent wrote:
> 
> I'm wondering if and how to make Mathematica evaluate  derivatives like
> 
> D[(1-x^2)y''[x],{x,n}]
> 
> y[x] is an unspecified function, n is a +ve integer.
> 
> It seems as though the system ought to "know" that this reduces to three
> terms only, provided that n is constrained?
> 
> -Phillip
> 
> ----------------------------------+----------------------------
>  Dr Phillip Kent                  | tel: +44 (0)171 594 8503
>  The METRIC Project               | fax: +44 (0)171 594 8517
>  Mathematics Department           |
>  Imperial College                 | p.kent at ic.ac.uk
>  London SW7 2BZ, U.K.             | http://metric.ma.ic.ac.uk/
> ----------------------------------+----------------------------
>  "Behaviour can be understood only as the history of behaviour"



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