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