Re: Change of variable in ODE's

*To*: mathgroup at christensen.cybernetics.net*Subject*: [mg598] Re: [mg585] Change of variable in ODE's*From*: Richard Mercer <richard at seuss.math.wright.edu>*Date*: Thu, 23 Mar 1995 10:10:20 -0500

> 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, First of all, both g and h produce the same output (looks like h) when evaluated, so it if you can do it for h, it will work for g too; just don't use Hold. For h, the following works. In[8]:= v[x_]:= u[x]/x^2 h /. y->v //Expand Out[9]= u''[x] This avoids the explicit use of pure functions and is relatively simple. It is not completely ideal in that it depends on y always occuring with an argument, i.e. y[x], but as long as you stick with that syntax you should be alright. Richard Mercer

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