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