Solving a single equation of two variables

Thank you all for your replies.  

Actually this curve is relatively well know.  It is the cartesion
representation of the tractrix curve used in various areas - my interest is
in its us as a horn loudspeaker (see
http://kosat.consultit.no/~ketil/lowther/bigfunht.html)  Wulfram has a polar
representation at http://mathworld.wolfram.com/Pseudosphere.html although
the Wulfram representation has the mouth pointing down and my cartesian
cordinates have it pointing to the side.  It is actually quite easy to graph
- Excel will do it easily and numerical solutions for r are quite easy (if
boring) to achieve using solver.

I was using it as a test case to increase my understanding of Maxima.  After
receiving your replies I looked at intermediate stages of the equation and
found it interesting how quickly Maxima breaks down in giving useful results
for non-polynomial equations.  

For example:

(C16) solve (x = log( sqrt( a^2 - r^2)),r);
			    		   2	   2 x		  2	 2 x
(D16) 	        [r = - SQRT(a  - %E   ), r = SQRT(a  - %E   )]
(C17) solve (x = log(a + sqrt( a^2 - r^2)), r);
	    x
Is  a - %E   positive, negative, or zero?

positive;			 		2    2      x
(D17) 			   [SQRT(a  - r ) = %E  - a]

Also:
	 		  
(C7) solve (x = log( sqrt( a^2 - r^2)),r);
			    	   2	   2 x		 2	 2 x
(D7) 	        [r = - SQRT(a  - %E   ), r = SQRT(a  - %E   )]
(C8) solve (x = log( sqrt( a^2 - r^2)/r),r);
				        2	 2    - x
(D8) 			   [r = SQRT(a  - r ) %E   ]
(C9) solve (x = log((a + sqrt( a^2 - r^2))/r),r);
				    	  2	 2	    - x
(D9) 		        [r = (SQRT(a  - r ) + a) %E   ]
(C10) solve (x = a* log((a + sqrt( a^2 - r^2))/r),r);
				   		2    2	  - x/a
(D10) 		       [r = (SQRT(a  - r ) + a) %E     ]

Best regards,
Herb

Herbert G. Desson, ACAS, MAAA

Actuary
JLT Risk Solutions
6 Crutched Friars
London EC3N 2PH

phone:  +44 (0)20 7528 4702
fax:       +44 (0)20 7558 3785


-----Original Message-----
From: Barton Willis [mailto:willisb at unk]
Sent: 16 September 2003 15:44
To: Herbert_Desson@jltgroup.com
Cc: maxima@math.utexas.edu; maxima-admin@math.utexas.edu
Subject: Re: [Maxima] Solving a single equation of two variables


I don't think you've done anything wrong; I don't think that you've found 
a Maxima bug either.
It seems to me that the solution to your equation is either a huge mess or 
it  doesn't have a reasonable 
representation.    (I think the second possibility is the correct one, but 
 I could be wrong.)  Neither 
commercial Macsyma or the Maple  (a somewhat out-of-date version)  give an 
explicit solution to your 
equation. 

Suggestion:  Rescale your equation with x -> a * x and r -> a * r.  You'll 
be able to eliminate a from your 
equation.  After that you can study your equation graphically and 
numerically.

Barton




Herbert_Desson@jltgroup.com
Sent by: maxima-admin@math.utexas.edu
09/16/2003 09:09 AM

 
        To:     maxima@math.utexas.edu
        cc: 
        Subject:        [Maxima] Solving a single equation of two variables


I am new to maxima and am trying to solve the following relatively simple
equation for r:
    x = a * log(( a + sqrt( a^2 - r^2))/r) - sqrt(a^2 - r^2),

where x and r are variables and a is a constant.



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