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"
- References:
- Way to evaluate D[(1-x^2)y''[x],{x,n}] ?
- From: Dr Phillip Kent <p.kent@ic.ac.uk>
- Way to evaluate D[(1-x^2)y''[x],{x,n}] ?