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