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Re: Change of variable in ODE's (fwd)
*To*: mathgroup at christensen.cybernetics.net
*Subject*: [mg606] Re: [mg585] Change of variable in ODE's (fwd)
*From*: mcdonald at delphi.umd.edu (William MacDonald)
*Date*: Mon, 27 Mar 95 17:02:35 EST
>From mcdonald Mon Mar 27 16:58:47 1995
>Subject: Re: [mg585] Change of variable in ODE's
>To: Jack.Goldberg at math.lsa.umich.edu (Jack Goldberg)
>Date: Mon, 27 Mar 95 16:58:47 EST
> Changes of variable in linear ODE's are used to convert an ODE
> into one of many normal or canonical forms. I thought it would be a
> simple task to get Mma to do the job for me. Well, yes and no.
> Consider the following two Mma versions of a particular
> Cauchy-Euler operator:
>
>
> (1) g = x^2 D[y[x],{x,2}] + 4 x D[y[x],x] + 2 y[x]
>
> and the more "natural" form
>
> (2) h = x^2 y''[x] + 4 x y'[x] + 2 y[x]
>
> The change of dependent variable y[x] = u[x]/x^2 (when appropriately
> effected) reduces (1) or (2) to the normal form u''[x]. The change of
> independent variable x = Log[t] reduces (1) or (2) to a constant
> coefficient 2nd order ODE. The catch is "appropriately effected".
> The following works for g.
>
> In: Hold[g]/.y[x_]->u[x]/x^2//ReleaseHold
>
> and
>
> In: Hold[g]/.y[x_]->y[Log[x]]//ReleaseHold
>
> It does not work for (2). After much labor I found an extremely
> awkward solution. Briefly:
>
> In: h/.Derivative[n_][y_][x]->Derivative[n][u[#]/#^2&][x]
>
> and
>
> In: h/.Derivative[n_][y_][x]->Derivative[n][y[Log[#]]&]][x]
>
> Actually, these don't quite work. These rules do not transform y[x]
> so a separate rule must be appended to change y[x] to u[x]/x^2 and
> y[x] to y[Log[x]]. I am particularly unhappy with this solution for
> these reasons: (1) They are awkward in the extreme. (2) They require
> the user to know the FullForm of y' and understand pure functions.
> Although I can't say that the method used on g is the best possible,
> it is easily understood by a student with a rudimentary knowledge of Mma.
> This is surely not the case with the rules used for h.
>
> Q1: Is there a way to effect the changes on h which requires less
> skill with Mma?
>
> Q2: Are there alternative ways to handle g?
>
> One last thought. I suppose one could write a package which contains
> the functions ChangeDependentVariable and ChangeIndependentVariable
> which would have the differential operator and the change of variable
> as arguments. Then all would be hidden from the user who would only
> need to call either of these commands.
>
> Q3: Is this the way to go?
>
> Jack
Jack,
How is this? To do the first transformation y->u/x^2:
In[78]:=
h[y_,x_]:=x^2 D[y,{x,2}] + 2 x D[y,x] + 2 y
In[80]:=
h[u/x^2,x]
Out[80]=
4 u
---
2
x
and to change the dependent variable:
In[82]:=
h[y,x]/.x->Log[t]
Out[82]=
2 y
--
William M. MacDonald
Professor of Physics
University of Maryland
Internet: mcdonald at delphi.umd.edu
--
William M. MacDonald
Professor of Physics
University of Maryland
Internet: mcdonald at delphi.umd.edu
--
William M. MacDonald
Professor of Physics
University of Maryland
Internet: mcdonald at delphi.umd.edu
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