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>