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MathGroup Archive 1995

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Transformation of homogeneous ODE

  • To: mathgroup at christensen.cybernetics.net
  • Subject: [mg610] Transformation of homogeneous ODE
  • From: mcdonald at delphi.umd.edu (William MacDonald)
  • Date: Mon, 27 Mar 95 18:01:15 EST

Apologies for a stupid response to Jack Goldberg's question.  I did
not check what was happening before firing off my answer, which was
about half-correct.

Write the ODE 
 h[y_,x_]:= x^2 D[y,{x,2}] + 4 x D[y,x] + 2 y
then evaluate
h[u[x]/x^2,x].
The answer is u''[x].  

Now to the second part, substituting for the independent variable,
    x = Log[t].
This does not turn this equation into one with constant coefficients.
I think Jack meant to use
    t = Log[x].
This should give the equation
     D[w[t],{t,2}] + 4 D[w[t],t] + 2 w[t] 
where 
        w[t_] := y[Exp[t]].
But this requires throwing in a factor of dt/dx at appropriate places.
I have not got this worked out.
-- 

William M. MacDonald
Professor of Physics
University of Maryland
Internet: mcdonald at delphi.umd.edu



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