NetNews Usenet Archive 1992 #27/NN_1992_27.iso/spool/sci/math/15264

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From: [e-mail] (Robert B. Israel)
Newsgroups: sci.math
Subject: Re: ODE problem...
Date: 19 Nov 92 23:08:20 GMT
Organization: The University of British Columbia
Lines: 59
Message-ID: <[e-mail]>
References: <[e-mail]> <[e-mail]>
NNTP-Posting-Host: unixg.ubc.ca

In <[e-mail]> [e-mail] (Bart E. Goddard) writes:


>following problem.  The solutions still works just fine, but one
>of the constants is different:


>In article <[e-mail]> [e-mail] (Robert
>B. Israel) writes:
>> >(x and y are functions of t, a is a constant)
>> >x'=(a)(x)cost+(a)(y)sint
>> >y'=(a)(x)sint-(a)(y)cost.
>>
>> Quite a neat problem!  I hope nobody was sadistic enough to
>> assign it as homework.
>>
>> First go to polar coordinates: x = r cos(s), y = r sin(s)
>> (I'm too lazy to type "theta").
>> (1)   r' = (x x' + y y')/r = r cos(t-2s)
>     there's a missing "a" here ^

>[....]
>> --
>> Robert Israel                            [e-mail]

>Bart Goddard

Oops, sorry about that.  So k has a missing factor of "a".  BTW, the
solutions can be written, without using arcsin, as
  x = exp(k t) (cos(t/2) + 2 (a-k) sin(t/2))
  y = exp(k t) (2 (k-a) cos(t/2) + sin(t/2))
where k = +- sqrt(a^2 - 1/4), for |a| > 1/2.

For |a| < 1/2, this still works but these solutions are complex.  Take
real and imaginary parts to get real solutions:
  x = (1/2-m) cos((1/2+m)t) + (1/2+m) cos((1/2-m)t)
                   + a sin((1/2+m)t) + a sin((1/2-m)t)
  y = - a cos((1/2+m)t) - a cos((1/2-m)t)
                   + (1/2-m) sin((1/2+m)t) + (1/2+m) sin((1/2-m)t)
and
  x = - a cos((1/2+m)t) + a cos((1/2-m)t)
                   + (1/2-m) sin((1/2+m)t) - (1/2+m) sin((1/2-m)t)
  y = -(1/2-m) cos((1/2+m)t) + (1/2+m) cos((1/2-m)t)
                   - a sin((1/2+m)t) + a sin((1/2-m)t)
where m = sqrt(1/4 - a^2).

For a = +-1/2, my formula only gives one solution:
  x = cos(t/2) + 2a sin(t/2)
  y = -2a cos(t/2) + sin(t/2)
A second linearly independent solution can be obtained using a limit
procedure:
  x = t cos(t/2) + (2at-2) sin(t/2)
  y = (2-2at) cos(t/2) + t sin(t/2)

--
Robert Israel                            [e-mail]
Department of Mathematics             or [e-mail]
University of British Columbia
Vancouver, BC, Canada V6T 1Y4