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Re: variable substitution in differential eqns
*To*: mathgroup at smc.vnet.net
*Subject*: [mg61975] Re: variable substitution in differential eqns
*From*: Peter Pein <petsie at dordos.net>
*Date*: Sun, 6 Nov 2005 05:12:30 -0500 (EST)
*References*: <dkhm55$2e8$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
Joseph Fagan schrieb:
> For Calculus of Variations, I need to make some hairy variable
> substitutions.
>
> A simple substitution example is shown here in the first few lines.
>
> See
> http://mathworld.wolfram.com/Second-OrderOrdinaryDifferentialEquation.html
>
> If I make the substitution x=1/z how can I get Mathematica to give
> me eqn (2)
> and eqn (5)
> and, being greedy, eqn (6)?
> or point me to where to begin.
>
> Thanks
> Joe
>
>
Hi Joe,
replace y[x] by y, y'[x] by Dt[y]/Dt[x] and so on:
In[1]:=
expr = y''[x] + p[x]*y'[x] + q[x]*y[x];
In[2]:=
Collect[Expand[Simplify[
expr /. {Derivative[n_][y][x]:>Nest[Dt[#1]/Dt[x]&, y, n], y[x] -> y}
/. x -> 1/z /. y -> y[z]]],
Derivative[_][y][z] ]
Out[2]=
q[1/z]*y[z] + (2*z^3 - z^2*p[1/z])*y'[z] + z^4*y''[z]
the missing handling of p and q can be done by
a) hand
b) more effort, writing a function.
but I hope this helps a little bit.
Peter
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