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Re: changing variables in differential equations

• To: mathgroup at smc.vnet.net
• Subject: [mg100647] Re: changing variables in differential equations
• From: dh <dh at metrohm.com>
• Date: Wed, 10 Jun 2009 05:34:10 -0400 (EDT)
• References: <h0l48t$opp$1@smc.vnet.net>

Hi Andre,

having clearified that v= V[u] is to be considered the new dependent 

variable there is still the question if the dependent variable remains 

essentially the same or not:

x=X[u,v[u]]

y=Y[u,v[u]]= V[u] = v



Anyway, we can define the i-derivative of y with respect to x by y[i] 

(note: lower case= variables, upper case= functions):

==========================

Clear[x,X,y,Y];

x[0] = X[u, v[u]];

y[0] = Y[u, v[u]];

y[i_] := Dt[y[i - 1]]/Dt[x[0]]

===========================

here is a simple example where the transformation only depends on u:

==========================

X[u_, v_] := u^2;

Y[u_, v_] := X[u, v] + X[u, v]^2

y[1]//Simplify

==========================

Daniel



Andre Hautot wrote:

> Hello,

> Given an ordinary differential equation of order n, H(x,y,y',y'',...)=0

> Given the change of variables : x=x(u,v) and y=y(u,v)

> How to calculate (hopefully recursively) y', y'', and so on as functions 

> of say : u, v', v''

> 

> In other words how to translate automatically the differential equation 

> in terms of the new variables ?

> Thanks in advance