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MathGroup Archive 1997

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Substitution of variables in differential equations

  • To: mathgroup at smc.vnet.net
  • Subject: [mg6294] Substitution of variables in differential equations
  • From: Michael Hartl <michael.hartl at uni-konstanz.de>
  • Date: Sat, 8 Mar 1997 00:26:16 -0500 (EST)
  • Organization: Univ. Konstanz
  • Sender: owner-wri-mathgroup at wolfram.com

How can I solve ordinary differential equations by the method of
substitution of variables? (I am using Mathematica 2.2.) 
In the following example I want to simplify an ODE (function y, variable
x, differentiation with respect to x) substituting x by t (with
x=Cot[t]):

In[1]:= dgl[y_] = Dt[y,{x,2}]+1/(1+x^2)^2 y == 0

            y
Out[1]= --------- + Dt[y, {x, 2}] == 0
              2 2
        (1 + x )

In[2]:= dgl[f[x]] /. x->Cot[t]

          f[Cot[t]]
Out[2]= -------------- + f''[Cot[t]] == 0
                   2 2
        (1 + Cot[t] )

In[3]:= % //Simplify

                        4
Out[3]= f[Cot[t]] Sin[t]  + f''[Cot[t]] == 0

The correct result should look similar to:

	Sin[t]^4 (y''[t] + 2 Cot[t] y'[t] + y[t])

Thank you for your help.

Mike

PS: In some textbooks on Mathematica (even ones dedicated to
differential equations) these steps were performed outside the
Mathematica sessions by hand!

-- 
michael.hartl at uni-konstanz.de


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