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