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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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