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From: albert@oliven.UUCP (LuigiAlberto Piodisavoia)
Newsgroups: net.math
Subject: Curious problem
Message-ID: <743@oliven.UUCP>
Date: Thu, 6-Dec-84 17:57:16 EST
Article-I.D.: oliven.743
Posted: Thu Dec  6 17:57:16 1984
Date-Received: Sun, 9-Dec-84 02:22:12 EST
Organization: Olivetti ATC, Cupertino, Ca
Lines: 57




While working on a theoretical physics problem I run into the
following mathematical curiosity:

Suppose you have two functions of the form

	y(x) = ax + b sin ( cx + d )

which would look something like this:


       |
       |
       |
       |
       |
       |                                           .
       |                                    ..    .
       |                             ..    .  . ..
       |                      ..    .  .. .    .. .
       |               ..    . ..  .  . ..    .    .
       |        ..  ...  .  .   ...  .    .  .      .
       |       .  .. ..   ..     .  .      ..        .
       |      .   ...  .  .       ..
       |     .   .      ..
       |        .
       |
-----------------------------------------------------------------------


The obvious possibilities that arise by manipulating the
coefficients are: 

i) the curves never intersect.

ii) the curves intersect a finite number of times.

iii) the curves intesect an infinite number of times.

I am interested in the second case, in other words:

QUESTION 1.

How many roots does

	ax + b sin ( cx + d ) = ex + f sin ( gx + h )

have when the coefficients a - h are such that the two curves
intersect a finite number of times ?  Express your answer in
terms of a-h.

QUESTION 2.

Can you guess what in what field of physics would such equations
arise ?