From: israel@math.ubc.ca (Robert Israel)
Subject: Re: analogue of 'resultants' for ODE systems?
Date: 25 Jul 1999 21:49:15 GMT
Newsgroups: sci.math
Keywords: eliminating variables

In article <379a01bf.28611943@news.pipeline.com>,  <nospam@127.0.0.1> wrote:
>Is there a generic method for eliminating variables from systems
>of ODE's analogous to the 'resultant' method for systems of
>algebraic equations?  I'm particularly interested in methods of
>eliminating time from systems of ODE's.

Consider, for example, a pair of equations x' = f(x,y,t), y' = g(x,y,t)
(where ' = d/dt).  Suppose you want to eliminate y.  Differentiate the
first equation and you have 
x" = f_1 x' + f_2 y' + f_3 = f_1 x' + f_2 g + f_3
If f and g are polynomials, you can eliminate y from these equations
by taking the resultant of x" - f_1 x' - f_2 g - f_3 and x' - f 
with respect to the variable y.  If they are rational functions, 
express  x" - f_1 x' - f_2 g - f_3 and x' - f as fractions and
take the resultant of the numerators.

On the other hand, if you want to eliminate t from this system, first make
x the independent variable:

dy/dx = y'/x' = g/f 
dt/dx = 1/x' = 1/f 

and proceed as above.

For example, eliminating t from the system x' = y, y' = t y + x, I get 
the equation

y^2 d^2y/dx^2 + x dy/dx - 2 y = 0

Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            
Vancouver, BC, Canada V6T 1Z2