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