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Re: Changing variables within a differential equation

  • To: mathgroup at smc.vnet.net
  • Subject: [mg105056] Re: Changing variables within a differential equation
  • From: dh <dh at metrohm.com>
  • Date: Thu, 19 Nov 2009 05:24:15 -0500 (EST)
  • References: <hdttaa$h4h$1@smc.vnet.net>


riccardo benini wrote:

> Dear all,

> 

> I've been looking for a while on the web for the solution of the following

> problem:

> 

> given a (partial) differential equation F(y''[x], y'[x], y[x], x)==0  (where

> y and x may be though as multidimensional),

> how can I correctly set up a change of variables y -> y[g],  x -> x[t],

> and get an equation in g[t]:   F2(g''[t], g'[t], g[t], t)==0?

> 

> 

> I think that this is an old problem for the Mathematica community (and

> probably settled down),

> but I can't figure out how to solve it and write a procedure able to do it.

> 

> Thanks in advance,

> Riccardo Benini

> 

> 

Hi Riccardo,

if I understand you correctly, you want to change:

y'',y',y,x ==>  g'',g',g,t

where:

let  us denote:

y[x]: y'=dy/dx,y''=d^2y/dx^2

g[t]: g'=dg/dt,g''=d^2d/dt^2

and the transformations: y= G[g], x=T[t], with G'=dG/dg, T'=dT/dt

With this we have:

y'= dy/dx= (G' g' dt)/(T' dt) = G' g' / T'

y''= d(G' g' / T')/dx= (G'' g'^2/T' + G' g''/T' - G'g'T''/T'^2)/T'

    = G'' g'^2/T'^2 + G' g''/T'^2 - G'g'T''/T'^3

If you replace y,y,'y'' in F by the above expressions, you get F2:

F2(g''[t], g'[t], g[t], t) = F(G'' g'^2/T'^2 + G' g''/T'^2 - 

G'g'T''/T'^3, G' g' / T', G, T)



Daniel




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