Date: Mon, 20 Mar 95 12:44:12 CST
From: rusin (Dave Rusin)
To: smib@uia.ac.be
Subject: Re: Need help with LINEAR ode (dear me)
Newsgroups: sci.math

In article <1995Mar17.091955.17948@reks.uia.ac.be> you write:
>Suppose I have the following two-dimensional system of ode's:
>
>\dot x = a x - y
>\dot y = x  + a y
>I want to follow the solution which goes through (-1,-a) (the orbit
>is tangent to the line x=-1 there) for another turn, until it
>hits the line x=-1 again. Question is: what is the y-coordinate
>at that moment?

You noted that what you need to do is solve an equation involving  arctan
and  ln.  Rather than exponentiate, take tangents: if I've done it right,
you need to solve
	-y = tan (  ln( (1+y^2)/(1+a^2) )/(2a) + arctan(a) ).
If you graph the function of  y  on the right side, you'll see a 
secant-like curve in the middle, with a low point a little below  (0,a),
and with asymptotes occuring where the term inside the tangent is +- pi/2.
(These asymptotes are a little outside  y = +- a ). Outside this curve,
the rest of the graph is a collection of tangent-like curves to the right
and cotangent-like curves to the left. In all cases, the location of
the asymptotes is easily found: the y-coordinates are the values of y
for which  ln((1+y^2)/(1+a^2))  is  (2a)*(-arctan(a) + odd multiples of pi/2)
(they are spaced exponentially far apart on the  y axis).

The line through the origin with a slope of  -1  is tangent to the first
part of this graph at the point (-a,a). You asked for the next point of
intersection, although I have lost track of the direction of time travel,
so I dn't knw if this is right or left. In either case, it is clear from
the graph that the next point of intersection is near the next asymptote,
particularly if  a  is large. For example, the next one to the left occurs
before the negative value of  y  having
	(1+y^2)/(1+a^2) = exp(-3a pi). exp( -2a arctan(a) ).

I'm not sure what kind of answer you seek. An asymptotic formula valid for
a  close to  0  or for very large  a  is probably possible. Do you
have any particular  a  in mind?

dave